We know from GR that an object of non-zero mass cannot propagate at light. I have a problem with the term speed of light. Does it mean the relative speed of that object with another object or frame of reference? Or the speed gained by constant acceleration from rest?
You are asking what your speed would be relative to or if it has to be relative to anything. Yes, motion of a massive body must always be relative to something.
Imagine you are in empty space (empty universe), moving at speed c/2. Now what is that speed relative to? Nothing. Would you know even you are moving? No.
This is not completely true, because we usually use the CMB as the relative reference frame of the universe.
You are confused, because sometimes people say, that there is no universal reference frame, yet, the speed of light is the absolute speed limit. OK, but the speed of light compared to what?
Now since we usually use the CMB as a reference frame for the universe, that is some kind of universal reference frame. Though, the absoluteness of the speed of light has nothing to do with that.
The speed of light in vacuum, when measured locally is always c. Relative to anything else (any other particle, object) in the universe. Two separate observers (massive) would see the same photon travel at the same speed c. It does not matter what speed the observers are traveling at, or what direction. Now you ask, but how can those two observers traveling at different speeds see the same photon travel at the same speed c? It is because you are looking at it the wrong way. You think you start at speed 0 in space and then you speed up.
Wrong. At the big bang, every particle was massless. All of them were traveling at the same speed c. It was the sea of photons. They were all traveling in the speed dimensions at speed c, and in the time dimension at speed 0. Time stops for the photons. They do not experience time as we (massive) do. Now, later as they (some particles) gained mass, they started to slow down in the spatial dimensions, and they started moving in the time dimension. They started experiencing time as we do. This is because the universe is built up so and the four velocity vector is built up so, that its magnitude is always constant, and for massive particles it is always c. Now if a particle slows down in the spatial dimensions, it has to speed up in the time dimension to keep the magnitude of the four velocity vector c.
Now every single particle in the universe is moving relative to the speed of light. For massless particles, this relative ratio is just 100%c. For massive particles it is less then 100%c. All particles are moving at speeds relative to this speed.
Now you are asking why the speed of light is relative, if it is the absolute speed. It is relative to any other observer (reference frame, massive observer) in the universe. What SR and GR are saying is that the speed of light (in vacuum, measured locally) is always c relative to any observer, any reference frame. Obviously, we tested this with experiments. Now all observers are massive, and they always do have a reference frame.
We do not know what the speed of light would seem relative to another object moving at speed c, because this question is meaningless anyway. The photon (or any massless particle, like gluons) does not have a reference frame. It does not make sense to ask what a photon would see.
But, the speed of light is c (in vacuum, when measure locally) to any observer. Since observers are massive, and do have a reference frame, our experiments showed exactly this. This means, that the speed of the observer is relative to the photon, and not the other way around.
This means, that the massive observer moves at speed 0.1c, the other observer moves at 0.01c, and both of them see the photon travel at speed c (in vacuum, when measured locally). You see this is no surprise now, that we established that both massive observers move relative to the speed of light. Of course they see the photon travel at the same speed.
This is of course only true for vacuum, and local measurements, because as per GR, the speed of light varies in a gravitational field (see the Shapiro effect).
I must say that your question is not very clear, so I will do what is not usually done: quote the OP.
We know from GR that an object of non-zero mass cannot propagate at light.
I guess you mean "at the speed of light". We know that from Special Relativity, on which GR is built.
I have a problem with the term speed of light. Does it mean the relative speed of that object with another object or frame of reference?
A speed always means "relative to a reference frame". You can calculate speeds of object relative to one another, but you actually go to one object's rest frame to do it in a meaningfull way. If you throw an object at 0.9c in one direction and another at 0.9c in the opposite direction, no physicist will ever say that the second object has a velocity of 1.8c relatively to the first.
And, because I guess this is your problem, lights travel at the same speed (in vacuum...) in any inertial frame.
Or the speed gained by constant acceleration from rest?
And this is the part I don't understand. :)