At first it seems suprising that this invariant is always the same for different objects and setups (that it is the same after Lorentz transformation is the meaning of invariant as you pointed out). So why is it for all objects equal to c.
If you think for stationary examples it just means that the eigenzeit has a different pace, so two clocks next to each other going at a different pace. But this is ruled out by another definition, such that clocks are references by light clocks. And now it is also clear why this invariance is c.
It's a unit vector, and physicists working with relativity habitually set C=1 to make the equations simpler. That being said, it isn't really a vector -- it's just a direction. The "equal to C" bit is a mathematical artifact of both vectors being normalized.
Because this 4-vector removes the factor of rest mass, which distinguishes objects. Putting back the rest mass converts 4-velocity to 4-momentum, which does not have the same magnitude for all objects--the magnitude is the rest mass.
If you think for stationary examples it just means that the eigenzeit has a different pace, so two clocks next to each other going at a different pace. But this is ruled out by another definition, such that clocks are references by light clocks. And now it is also clear why this invariance is c.