Why are observers/reference frames able to see themselves moving through time but not through space? All observers are stationary in their own reference frames.
That is, their space coordinates are constant at all times (in their frame).
However, they can see themselves moving through time.
What is the fundamental difference between the space axes and the time axis that enables this peculiar behavior?
 A: Particles have proper time, they are aging, and this is an intrinsic process of the particle. 
So, if you are enclosed in a ship without windows, you will not know where you are moving, but you will know that (your time) is passing, and that you are aging.
Time and space are different things, their similarity is limited to Lorentz symmetry.
A: Basically you are asking why you cannot see yourself stop moving through time. To answer that, we need to distinguish between time and flow of time.

To distinguish between events happening at the same point in space but at different times we need to specify when an event happened as well as where it happened, so we add a time coordinate t. Events can then be uniquely located by their spacetime coordinates (t,x,y,z). To a physicist time is just a coordinate used to specify events in spacetime.
  So spatial coordinates are observer dependent. However time is absolute. Assuming we both use Greenwich Mean Time (or some other similar standard) we will always both agree on the time no matter where we are on Earth or however we are moving relative to each other. In Newtonian mechanics time is special for this reason, so it makes sense to consider it separately from space.
  The point of all this is that in relativity time is not uniquely defined. When we consider the coordinates used by different observers we find that time and space get mixed up with each other. Time is no longer distinct from space, and that’s why physicists treat it as just one of the four coordinates that together make up four dimensional spacetime.

https://physics.stackexchange.com/a/235512/132371
Now why does time flow (in the everyday sense)? The answer is entropy. Entropy is an extensive property of a thermodynamic system. It is related to the microstates that a macroscopic system can have, and is consistent with the macroscopic quantities a system can have. The second law of thermodynamics states that the entropy of an isolated system never decreases over time. Isolated systems spontaneously evolve towards thermodynamic equilibrium, maximum entropy.
https://en.wikipedia.org/wiki/Entropy

Our scientific time definition uses the concept of entropy to codify change in space, and entropy tells us that there exists an arrow of time.
  In special relativity and general relativity time is defined as a fourth coordinate on par with the three space directions, with an extension to imaginary numbers for the mathematical transformations involved. The successful description of nature, particularly by special relativity, confirms the use of time as a coordinate on par with the space coordinates.
  It is the arrow of time that distinguishes it in behavior from the other coordinates as far as the theoretical description of nature goes.

https://physics.stackexchange.com/a/32074/132371
Now basically in our own reference frame, we cannot stop the flow of time, because we do have rest mass. It does not make sense to say things like what would we see from the reference frame of a photon (massless). Because massless particles do not have a reference frame. Sometimes you hear phrases like photons do not experience time or photons see time stop. In reality, photons do not have a proper time. And this has to do with them being massless. As long as you are in a reference frame of an object that does have rest mass, you cannot stop the flow of time.
How to define the proper time of a photon?
So basically massless particles are the only ones that do not have a proper time,  but since your question is about objects (you do not specify it, but I assume with rest mass), you cannot stop the flow of time in your frame. To do that (to experience time differently) you would have to lose rest mass.
You are basically asking what makes the temporal dimension different from the spatial ones, and the answer is the arrow of time.
A: When you drive to a hardware store, you want to take your ruler with you. Leaving it at home is not helpful. We move in time and want to take our coordinate systems with us instead of leaving them behind in the mist of time where we couldn't use them.
A coordinate frame not moving in time can be easily constructed. Let's call it "coordinate frame" instead of "reference frame", because "reference frame" typically means "coordinates of a physical observer moving in time". A coordinate frame not moving in time would exist only for an instant. For example, you need to measure the length of a flexible object at a particular moment of time.
You attach a ruler to the object and a clock on each side of the object. You synchronize the clocks in advance. When both clocks show the same time, two measurements are made using the ruler on two sides of the object. The difference between the measurements represents the length of the object at the exact time of the measurement (as long as the ends of the object do not move relativistically fast).
The measurement we just made was made in a coordinate frame that existed only at the instant of the measurement. The time coordinate was the same everywhere, but the space coordinate was variable. Thus we have constructed a coordinate frame not moving in time.
While such frames are possible and sometimes can be used to measure spacelike intervals, reference frames that move with us in time are more convenient in mechanics, relativity, and cosmology to measure timelike intervals or proper time of physical objects.
