I understand the distinction between phonemes and allophones, but why are phonemes considered as mental objects and not psychological objects? Isn’t everything mental also, in a sense, necessarily psychological? I wondered if there is a point to splitting mental from psychological stuff.
Phonemes are "mental objects" in the sense that they exist only in the mind: no matter how many recordings you make, and how many formants you analyze, you'll never find some distinctive label on a spectrogram that indicates "this is a phoneme, and this isn't".
You could equally well say that, since they only exist in the mind, they're "psychological objects". But just like with different phonemes, the difference between "mental objects" and "psychological objects" is only meaningful in context, and I've never seen an important distinction made between the two.
For your final question, "if there is a point to splitting mental from psychological stuff", I would say no. I've never seen such a distinction made in linguistics.