Indeed, it's not even the same between pieces!

Kings have Chebysev geometry while Rooks have taxicab geometry: https://en.wikipedia.org/wiki/Taxicab_geometry#See_also

It's left as an exercise for the reader to figure out the geometry of the remaining pieces.

Rooks don't have taxicab geometry. Their metric space is compact even on an infinite board. I think you're thinking of the wazir: https://en.wikipedia.org/wiki/Wazir_(chess)

https://en.wikipedia.org/wiki/Chebyshev_distance for kings, but on a clear board the rook distance between any two squares is 1 or 2, whereas with Taxicab it could be as much as 14.