When physicists talk about non-Euclidean geometry in relativity, they usual mean gravitational masses curving space. If not that, they mean some clever formulas that relate hyperbolic space to velocity addition formulas and other aspects of special relativity. I mean something different, and more basic.
Euclidean geometry means 3-dimensional space (or Rn) along with the metric given by the Pythagorean Theorem, and the symmetries given by rotations, translations, and reflections.
Other geometries are defined by some space with some metric-like structure, and some symmetry group.
Special relativity is spacetime with the metric dx2 + dy2 + dz2 - c2 dt2, and the Lorentz group of symmetries. It is called the Poincare group, if translations are included.
That's it. That's why the speed of light appears constant, why nothing can go faster, why we see a FitzGerald contraction, why times are dilated, why simultaneity is tricky, and everything else. They are all byproducts of living in a non-euclidean geometry.
I am not even referring to curved space, or hyperbolic space. I mean the minus sign in the metric that makes time so different from space. It gives flat spacetime a non-euclidean geometry.
Historically, H. Lorentz viewed relativity as an electromagnetism theory. He viewed everything as based on electromagnetism, so I am not sure he would have distinguished between a spacetime theory and an electromagnetism theory.
Now we view electromagnetism as just one of the four fundamental forces, but Lorentz was not far off. At the time, it was not known that electromagnetism underlies all of chemistry, so he was right to view it as much more pervasive that was commonly accepted.
Poincare was the first to declare relativity a spacetime theory, and to say it applies to electromagnetism or gravity or anything else. He was also the first to write the spacetime metric, and the Lorentz symmetry group. He did not explicitly say that it was a geometry, but that would have been obvious to mathematicians at the time.
I credit J.C. Maxwell with being the father of relativity. He was till alive in 1972 when the Erlangen Program for studying non-euclidean geometries was announced. He probably had no idea that it would provide the answer to what had been puzzling him most about electromagnetism.
Minkowski cited Poincare, and much more explicitly treated relativity as a non-euclidean geometry. He soon died, and his ideas caught on and were pursued by others.
Einstein understood in his famous 1905 paper that the inverse of a Lorentz transformation is another Lorentz transformation, and that the transformations can be applied to the kinematics of moving objects. Whether he got any of this from Poincare is hard to say. Einstein rejected the geometry view as the basis for relativity. He did partially adopt Minkowski's metric about 1913, but continued to reject relativity geometrization at least until 1925. Carlos Rovelli continues to reject it.
The geometry is what enables relativity to explain causality. It is what made J.C. Maxwell's electromagnetism the first relativistic theory, and hence the first truly causal theory. Everything is caused by past events in the light cone. The non-euclidean geometry restricts the causality that way.
Science is all about reductionism, ie, reducing observables to simpler components. Applied to space and time, it means locality. Objects only depend on nearby objects, and not distant ones. Events depend on recent events, and not those in the distant past. It is the Euclidean distance that defines what objects are nearby. Clocks define the recent past, but is an event recent if it is a nearby object at a recent time?
Maybe yes, maybe no. That is where we need the non-euclidean geometry. The Minkowski metric is a fancy way of saying that the event was recent if light could have made the trip quickly.
If spacetime had a Euclidean geometry, then two events would be close if they are close in space and close in time. Causality does not work that way.
An event might seem to be spatially close, but if it is outside the light cone, then it cannot be seen, and it can have no causal effect. That is the consequence of the geometry.
Special relativity is usually taught based on the Michelson-Morley experiment on the motion of the Earth, and that is how it was historically discovered. It became widely accepted after Minkowski convinced everyone in 1908 that it was all geometry.
Today general relativity books emphasize the non-euclidean geometry of curved space, but the textbooks do not explain that the non-euclidean geometry of flat spacetime is at the very core of special and general relativity.