That seems very neutral to me? I'm struggling to think of a term that's not worse ("allowance") or less accurate.

[EDIT] It just means money that's expected to be spent, but not budgeted specifically for anything, and is largely put to personal discretion without any kind of accounting expectation—no? Some phrase using "discretionary" might also work, but that one seems too technical or formal.

Seems this is US-only usage, maybe it's not universal there, for me in the UK pocket money has all the connotations of childhood, I'd only use it to be deliberately patronising, it's like a full reversal where allowance makes me think of a meal allowance from work etc.

I think you're confusing that with "pocket change" as in a small/insignificant amount of money. Pocket money can just mean that you always have money around that can be spent according to whatever whim or whatever you want to do.