These games only explore allowing overlapping 3-dimensional spaces with clearly defined boundaries between those overlapping 3D spaces.
Higher dimensionality is largely irrelevant to whether the geometry is euclidean or not.
Hyperbolic geometry, on the other hand, is alien to most people, (especially fractal like the projections into 2d and 3d), so when people wave their hands and say 'non-Eyclidean geometry!', that's what they usually actually mean.
Try actually watching some of the hyper-rogue gameplay: https://www.youtube.com/watch?v=lX5eCfRSCKY
I assure you, it's more interesting as a geometric visualisation than planetary annihilation is.
More technical reading here: https://math.stackexchange.com/questions/1407550/what-hyperb...