Relativity just saws that certain physical quantities should remain invariant regardless of the reference frame they are viewed in.

A Vector is invariant regardless of the frame it is viewed in – even though it’s individual components are different in different frames.

A Tensor is also  an invariant.

The Space Time Interval in Special Relativity is the invariant that points a way out of all relativity paradoxes.

However, it still behooves to ask, Why The Spacetime interval and not some other quantity?

And for WAVES (see wave characteristics), what aspects of a wave can we call invariants (Phase? Phase Velocity? Group Velocity?). The PHASE of a wave is the number of wavefronts that pass through a window in a given time interval. This is an invariant.

Relativity of Motion Existed even prior to Einstein

Relative motion helped explain a lot of mechanical phenomenon. It just failed to explain electromagnetic phenomenon, notably, the lack of any standing wave solutions for light waves.

This is what led Einstein down the path of his special theory.