What is meant by coordinate time? Isn't it time measured on a clock? If not then what does it measure? In this question-
Is proper time equal to the Invariant Interval or the time elapsed in the Rest Frame?
@Dale in the comments says-

no $dt$ is never physical time. It is always coordinate time. The equation $d\tau=dt’$ needs to be interpreted carefully. It is not a general fact, but instead it is only true along the worldline of the clock and only in the (primed) rest frame of said clock. But even then, the physical time is $d\tau$ and the coordinate time $dt’$ is designed to match it. That is what makes the primed coordinates the rest frame of the clock [...]

What is coordinate time?
What does one mean by coordinate time? Isn't it a time measured on the clock ( any clock)?
@Dale says no.
Then what is meant coordinate time?
 A: In relativity we deal with spacetime which is a 4-dimensional structure called a (pseudo-Riemannian) manifold. It includes space and time together. Meaning that time is just another direction in spacetime, perpendicular to the three dimensions of space.
In the manifold we have a metric which describes all of the geometric properties of spacetime. It describes physically measured distances and durations and angles and speeds. You can use it to identify straight or curved lines and so forth. Straight lines in this sense are technically called geodesics.
A point particle forms a line in the spacetime manifold, called its worldline. An extended object forms a worldsheet or a worldtube, but we will stick with worldlines for now. A physically measured duration is called proper time, $d\tau$, and is measured by a physical clock on a specified worldline. An inertial object (in free-fall, no real forces acting on it) moves in a straight line at constant speed according to Newton’s first law. In other words, an inertial object’s worldline is straight.
Now, spacetime does not come with any physical labels or grid lines on it, but often we find it convenient to add labels/gridlines. These are called coordinates. Coordinates are ordered set of four numbers that label each event in spacetime. They must have some nice mathematical properties like being smooth and invertible. Everything else is a matter of convenience.
One approach that is often convenient is to use the first coordinate for time and the last three coordinates for space $(t,x,y,z)$. A further convenience is to use straight lines and planes. A further convenience is to choose an inertial object as a reference and orient those lines so that the time coordinate axis is parallel with the reference object’s worldline and so that the space axes remain fixed and don’t rotate. A final convenience is to scale the coordinates so that on the reference object’s worldline the physically measured proper time matches the coordinate time and so that coordinate distances match measured physical distances. Such coordinates are called the inertial reference object’s rest frame. In the absence of gravity such coordinates can extend to cover all of spacetime, but with gravity they can only be local, near the path of the inertial object.
Importantly, in these coordinates the metric can be written as $ds^2 = -c^2 d\tau^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$. The physical time measured on any clock can be obtained by integrating the above expression for $d\tau$ along the worldline of that clock. The coordinate time is merely a convention for assigning the labels as described above. This convention is chosen to match the physical time on the reference object’s clock, but not necessarily on other clocks. Elsewhere, $dt$ is merely a label.
A: In special relativity one traditionally imagines that spacetime is filled with a network of clocks that are Einstein synchronized. The clocks define an inertial reference frame, and the time coordinate of an event is the reading on the nearest clock.
When dealing with curved spacetime you usually can't Einstein synchronize clocks, and it's usually not convenient to use clock readings (however synchronized) directly as a coordinate, so coordinates are treated more abstractly. The same is true when dealing with ordinary curved surfaces like the surface of the Earth. We use latitude and longitude to measure global position, but there are no rulers that measure latitude and longitude; measurements are done locally with metersticks. In general relativity, the metric tells you how to convert between local measurements of clocks and metersticks and the global coordinates that you're using.
A: Mathematically Space-Time is a 4-dimensional Lorentzian manifold $L$. Coordinates of an event,
i.e. $(x_{\mu})=(ct, x, y, z)$ only make sense when you define an Atlas $\mathcal{A}={(\mathcal{U}_{\alpha}, \phi_{\alpha})}$ on $L$ where
$\mathcal{U}_{\alpha}\subset L$ and $\phi_{\alpha}:L\rightarrow\mathbb{R}^4$  is a homeomorphism (i.e. continus functions between the topological Spaces $M$ and $\mathbb{R}^4$). The coordinates of a point $p\in U_{\alpha}\subset L$ are then given by $\phi_{\alpha}(p)=(x_1, x_2, x_3, x_4)$.
There are a few restrictions on the $\phi_{\alpha}$ in order for the math to make sense, but other than that you're more or less free to choose any parametrization you want. The main idea is now, that physical quantities and laws should in a sense be independent of this choice. $\tau$ is such a quantity, the coordinates $x_i=\phi_{\alpha}(p)_i$ are obviously not.
A: $d\tau$ is an invariant of the metric. $dt$ is the time measured by a clock in a particular frame, as $dx$ is the length measured in this frame.
It is like saying that in 2D spatial cartesian coordinates, $ds^2 = dx^2 + dy^2$. $ds$ here is invariant by rotation. And we can also say that for any orientation of $ds$, there is a particular frame so that $ds^2 = dx^2$ (or $ds^2 = dy^2$).
But for 2D polar coordinates it is different. $ds^2 = dr^2 + r^2 d\theta^2$. $ds$ is also invariant by rotation. But there is no angle of rotation that leads to $ds^2 = d\theta^2$. It only happens in the case of $r = 1$. It corresponds to the example of $d\tau = dt$ for a static observer in the Schwartschild metric, that is only valid at $r = \infty$.
The Minkowski metric can be compared to the example of cartesian coordinates, and $d\tau$ can be understood as the time for a clock at rest of an inertial frame.
But in GR, similar to the example of polar coordinates, it is not generally true.
