Height and normal detail come from the Bump or Displacement nodes in your material. In Blender 5.x the Bump node has a very small default Distance, so raise Distance (for example to 0.02) and bake again. For Height, the material needs a Displacement or Bump node, and the panel warns you if it is missing.

Any engine that uses image textures: Unity (URP and HDRP packs included), Unreal Engine (ORM pack and DirectX normals), Godot and anything that reads glTF. You can also export the model as GLB, glTF or FBX with the baked textures, or use the Substance Painter preset to bake mesh maps.