Using "standard chess rules" (like in several answers to the linked problem)
Already covered by other answers - 56 are possible, 28 of each colour
All other pieces could use the solutions using colour-agnostic pieces that attack every other piece, as shown in several other answers, but changing one of each pair to black.
e.g. 20 bishops and 32 knights (10 of each colour and 16 of each colour respectively)
The remaining solutions below all make use of the fact that same-colour pieces are NOT attacking each other.
64 are possible, 32 of each colour
Kings - the answer under standard chess rules would be zero, as a king cannot attack another king... but that's boring, so
if we modify the rules to allow multiple kings, and to allow kings to attack kings of an opposing colour (but kings of the same colour would be "protecting" each other and therefore not attacking), this makes 32 (16 of each colour) easily possible, for example
With same-colour knights treated as not "attacking" each other, we can place 48 (24 of each colour) as follows:
The left-hand one builds on the known solution for 32 colour-agnostic knights, by arranging the colours in patterns that allow two more similar blocks of 8 to be added. The right-hand one arranges 6 blocks of 8 in a ring around the edge of the board.
52 are possible, 26 of each colour.
(this looks a lot less complicated when you isolate only the dark or white squares...)