Your hunch is correct. The games' maps were built on square blocks of different heights and plane angles. Tr3 used triangle based maps, but still not true polygonal builds.

Tomb Raider was developed for the Saturn. Saturn used only quads. Saturn's dead by 1999. You can figure this out.

No need to get condescending. I was wondering about the cube-based geometry more than anything.

I suppose it also has to do with the limited technology of the time. Tomb Raider is essentially a platformer, and one of the first 3d platformers at that. If you compare it to eg Mario or Prince of Persia, those have the same 'fixed distance' approach, but in 2d.
A regular 3d grid just makes a lot of calculations a lot easier, as far as jump-distance, aligning to ridges etc goes.
Generic polygon meshes would require far more advanced raycasting, and I suppose the memory and CPU requirements just weren't there yet, on the intended target platforms.

Indeed. For example, a running jump requires exactly one "cube" to achieve. These calculated and deliberate jumps and animations is why the original games feel very similar to Prince of Persia.

The organic environments in tr 1 are pretty impressive though given the limitations. Maps were laid on a grid but the grids were also limited in size, so the environments are just a series of small gridded rooms connected together, with I think some rules on how they should and shouldn't join.

Tomb Raider 1&2&3&4&5's engine

you can correct the topic 😉