[KK7NIL]

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.

[daynthelife]

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)

[jibal]

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.