Red and brown mushrooms can't be placed on top slabs or stairs

probably intended feature 

I don't think this is intended. Most other things like torches only require to be placed on top of a solid face. The same should be true for mushrooms as well.

Can confirm in 20w51a.

Can confirm in 21w05b.

Can confirm in 21w06a.

Can confirm in 1.19.

Can confirm in 1.19.2.

Can confirm in 23w31a.

Code Analysis:

^{(Decompiled with MCP)}
Currently, the applicable blocks for placement are ones that are "isSolidRender" (Full cube blocks)

protected boolean mayPlaceOn(BlockState blockState, BlockGetter blockGetter, BlockPos blockPos) {
      return blockState.isSolidRender(blockGetter, blockPos);
   }

Possible Fix:

One fix for this, is to use "isFaceSturdy" instead, which only checks for a certain face of the block being applicable for placement (In this case, the top face).

protected boolean mayPlaceOn(BlockState blockState, BlockGetter blockGetter, BlockPos blockPos) {
      return blockState.isFaceSturdy(blockGetter, blockPos, Direction.UP);
   }

This provides the following behavior:

Can confirm in 1.20.6 and 1.21 Release Candidate 1.