Why is centrifugal force called fictitious? When an object undergoes rotation, from the object's reference frame, which is a non-inertial reference frame, the object feels there is a radially outward force, a centrifugal force, acting on it. However, from an inertial reference frame, this force doesn't exist at all. That's why it is called a fictitious force.
My argument is, who are we to say what is fictitious or not. The object at the non-inertial frame really feels the centrifugal force! So, it is a real force for the object.
Suppose, there are two inertial reference frames $S$ & $S'$ and $S'$ is moving with a velocity v that is a significant fraction of the speed of light. From $S$ it would seem that time is going slower for $S'$. Surprisingly, it would seem from $S'$ that time is going slower for $S$ as well. Now, who is right? Answer: Both of them are right.
So, is it really right to call centrifugal force fictitious just because it doesn't exist in an inertial reference frame?
 A: Without arguing why something is called something, I'll just give the justification for calling something, something.

Whenever a force emerges due to our choice of a non-inertial frame, it's called a fictitious force.
Consider as an example a person on an upward accelerating lift. If the person relies on his own frame, then he should write the equation of motion as
$$-mg+N=ma=0$$
But this is wrong! Because the person is in a non-inertial frame. He must take this into account. There are two ways of doing this. First, he could analyze the problem from an inertial frame. In which case, he would write
$$-mg+N=ma_\text{lift}$$
Or, he could introduce a force for which he has no reason. It's not from some sort of interaction of bodies such as gravitational or electrostatic forces. It's just a force out of nowhere. We call it a fictitious force. We write:
$$-mg+N-ma_\text{lift}=0$$
Now, this is exactly the same as what is done in the initial frame.

In exactly this sense, the centrifugal force is fictitious.
In an inertial frame, we write
$$m(\ddot{r}-r\dot{\theta}^2)=F_r$$
$$m(r\ddot{\theta}+2\dot{r}\dot{\theta})=F_\theta$$
In a non-inertial frame, we write
$$m\ddot{r}=F_r+F_\text{fic}=F_r+mr\dot{\theta}^2$$
$$mr\ddot{\theta}=F_\theta-2m\dot{r}\dot{\theta}$$
If we are writing the second, we have no explanation for the fictitious term.
A: I disagree that you feel centrifugal force. A person in a centrifuge actually feels their reaction to the centripetal force. If you sit in a car that is subject to harsh acceleration, you 'feel' as if you are being pushed back in your seat. There is no force pushing you back- it is simply the result of your inertia.
A: A force is fictitious if it doesn't obey Newton's three laws of motion. Recall Newton's first law:

The first law states that an object at rest will stay at rest, and an
object in motion will stay in motion unless acted on by a net external
force.
Mathematically, this is equivalent to saying that if the net
force on an object is zero, then the velocity of the object is
constant.
$$\sum \mathbf F=0\iff \frac{d\mathbf v}{dt}=0$$

You could call the first law a 'test' for the remaining two laws; if the first law doesn't hold you can't apply the other two. For example in a rotating reference frame an inertial path could look like the left picture:

The ball is obviously accelerating but there doesn't seem to be anything causing the force. You have two options in this situation. You could argue there is no force on the ball (I don't see anything causing a force!) and in that case Newton's first law would be violated because the ball is still accelerating. You could also argue that there is a force on the ball defined by $\mathbf F\equiv m\, \mathbf a$. In that case Newton's third law would be violated because there is no second force that is equal and opposite to $\mathbf F$. This violates momentum conservation and we have no choice but to call this force fictitious. Either way we conclude our reference frame is non-inertial.
Fictitious forces have plagued classical physics for a long time and they have only been resolved after the introduction of general relativity.
A: According to the  basic Newtonian formulation of mechanics, "real" forces come in couples: a force (action) and its reaction acting on the source of the action. Furthermore "real" forces are independent of the used reference frame. Fictitious forces, as centrifugal or Coriolis one, violate both conditions. This is the reason why they are called in that way. Maybe that is not a good name because they have concrete effects (though the interpretation of these effects may be given without  using fictitious forces by  just describing the phenomenon in an inertial reference frame).
A: 
The object at the non-inertial frame really feels the centrifugal force! So, it is a real force for the object.

This is actually incorrect. An accelerometer mounted on the object detects only the sum of the real forces. There is no experiment by which you can “feel” any fictitious force. They only exist in non-inertial frames and their existence or non-existence does not change the result of any physical measurement.
The distinction between inertial forces and real forces is physical, not merely a labeling reflecting some human prejudice.
A: "Fictitious" is a loaded term. It is better to use the alternative terms "pseudo force" or "inertial force", which avoid implying that the force is not real. Whichever label is used the key point is that centrifugal force (and, similarly, Euler force and Coriolis force) arises purely from the fact that we are measuring the position, velocity and acceleration of the object relative to a non-inertial reference frame.
A: 
So, is it really right to call centrifugal force fictitious just because it doesn't exist in an inertial reference frame?

Fictitious forces only exist in non-inertial frames.
Reactive centrifugal force (or whatever term you want to call it) is real and part of a pair of Newton third law forces. Consider a puck sliding on a frictionless surface, being centripetally accelerated in a circular path by an attached string that goes through a hole in the center of the circular path. The string exerts a centripetal force on the puck, coexistent with the puck exerting a reactive centrifugal force on the string.
As for the person in a centrifuge example, imagine the person is holding a brick against their stomach. The person will feel the reactive centrifugal force from the brick, coexistent with the centripetal force that the person exerts onto the brick.
https://en.wikipedia.org/wiki/Reactive_centrifugal_force
A: 
the object feels there is a radially outward force

No, it does not, and this is the whole reason behind the misconception.
A human may feel this way, but this is a thing that goes on in your head, and simply means that we are not wired in a way that we can really process the forces at play in a meaningful way on a biological level - mostly because it never happens in real life, so there is no selector for evolution to develop it.
Tie a rope to a large rock and twirl it around you (in free space with no planet close by, to avoid any confusion with its gravity). Cut the rope with one fell swoop. In this moment, no more forces act on the rock. The rock does not fly radially away from you; it flies away tangentially to the circle it just described. There was no real force pointing away from you.
You can easily test this in your garden, or from your armchair by watching a hammer throw in slow motion. Note how the athletes keep rotating a quarter turn after letting go, before following the flying object with their eyes.
While the rock was still bound, there acted exactly one force on it: the force of the rope pulling straight towards you; or from the perspective of the rock, towards its "bottom" (i.e. the centripetal force - from latin "pedes: on foot", like in "pedestrian"). While rotating around you, at each moment it had a speed tangential to its circle or orbit; and the centripetal force is the one that makes it "fall" towards you, or in more neutral terms, constantly changes its direction.
In the case of a solid rotating object, the rope is replaced by the electromagnetic forces that keep the particles together. In the case of an object standing on a planet, the rope is replaced by gravity. Oh, and a fun thought experiment: if someone would switch off gravity right now, on Earth, and a totally inelastic object were sitting on a completely inelastic floor (i.e., with no internal "spring" forces in the picture), and maybe in a local vacuum, it would seem to start flying parallel to the floor, not towards the sky.
The illusion of the centrifugal force is common to us and created by our brain; for example if you sit in a car and go fast around a corner, your head seems to move towards the side of the car. This is not the case though. Your head wants to go straight ahead; the side of your car is moving centripetally and thus pushing on your head if you relax your muscles enough that your head rests against the side.
A: "The object at the non-inertial frame really feels the centrifugal force! So, it is a real force for the object."
Three points on this:

*

*This is a philosophical observation, not physical.  Which does not mean it is invalid.  It's just that all kinds of mischief arise when people confuse philosophical statements with physical statements.

*In physics, a force is what causes an acceleration.  Period.  An object, person, cat, whatever, being rotated in some container about an axis is by definition accelerated radially inward.  So from the point of view of what causes of acceleration are present, the force is also radially inward and there is no radially outward force.

*But philosophically, I would argue that you don't feel any outward force.  Think about it.  If you are in a car making a sharp left turn and you wind up leaning with your right side against the car door, are you feeling any force on your left shoulder pushing you out?  No.  The only force you are actually feeling is on your right shoulder and it is pushing you in.  The "feeling" of an outward force is a psychological interpretation of your experience that integrates a whole host of perceptions in that moment.  The only force you really feel is inward.

A: Imagine standing on a train where the floor is made of ice. The train is parked at the station. You are standing on the ice floor of the train, holding onto nothing. There are other passengers seated on the train.
As the train pulls out of the station, you stay in place inertially, because the ice provides no shear force to your feet due to a lack of friction. To a passenger sitting down, he looks at you accelerating towards the rear of the train and remarks, "Wow, there is a lot of force acting on that guy!" Upon close examination, however, the passenger notes the ice floor and realizes there is no such force to explain your rearward acceleration. The passenger concludes that this apparent force is fictitious and that they are instead simply making observations from a non-inertial frame.
A: 
From S it would seem that time is going slower for S′. Surprisingly, it would seem from S′ that time is going slower for S as well. Now, who is right? Answer: Both of them are right.
So, is it really right to call centrifugal force fictitious just because it doesn't exist in an inertial reference frame?

S and S' don't disagree on how fast time is passing. They both agree on how much of S-time passes, and they both disagree on how much S'-time passes. What they disagree on is that S calls S-time just "time", while for S', the term "time" refers to S'-time. This isn't really a disagreement about a physical phenomenon such as time, this is merely a semantic disagreement about what physical phenomenon gets the label "time".
According to Einstein's Special Relativity postulates, neither of their labeling systems are superior. They both work equally well.
In the case of inertial versus non-inertial frame, it's still not quite correct to say that one is "right". Both of them have valid coordinate systems, and they have physical laws that predict phenomena. However, the non-inertial physics is more complicated.
Since it's good to have a universal set a of physical rules rather than having different rules depending on what frame of reference one chooses, and there is an objective basis on which we can say that the inertial frame, while not necessarily more "right", is more useful, it makes sense to formulate physics laws in terms of inertial frames, and that means having the word "force" refer to forces that exist in inertial frames, and adding the qualifier "fictitious" to phenomena that appear only in non-inertial frames.
A: The answer depends on the particular meaning that one puts in word fictitious:

*

*On the one hand, there is no doubt that centrifugal force (and other forces arising in non-inertial reference frames) is real - in the sense that it is measurable and has real physical effects (in the reference frame of the rotating object)

*On the other hand, real forces are characterized not only by their action, but also by their nature - the four fundamental forces (gravity, electromagnetic interaction, weak and strong interactions) reflect properties of the objects which are independent (or at least invariant in respect to) the reference frames that we use. Centrifugal force does depend on the reference frame for its existence, even if considering inertial reference frames superior to the non-inertial ones coudl be considered a matter of convention/existence. It is in this sense that the centrifugal force is considered fictitious. (Note that real forces themselves may also undergo drastic transformations when changing a reference frame, even when between the inertial reference frames.)

*Finally, let me point out that the whole of physics is fictitious - existing only in the human minds - as opposed to the actual nature, which behaves the way it behaves without any knowledge of how we, humans, describe and interpret it. In this sense, what we call fictitious or real is a matter of convention - and we have decided that the inertial frames of reference are superior (and let us not forget that some people were prosecuted by inquisition allegedly for promoting this view.)

Remarks:

*

*My last remark about inquisition is more grounded in popular beliefs than in facts. Galileo and Giordano Bruno were famously prosecuted by the inquisition (the first was forced to renounce his views, the second was burned at stake). The deeper look into both cases however shows that the true reasons for prosecution had little or nothing to do with science.

