Depends on jurisdiction, obviously, but not always the case. Technically in New York any lost property with a value over $20 must be turned into police (NY Personal Property Law §§ 251–258). But as you say the ability to prove ownership of loose currency is basically nonexistent (unless you’re Mr burns in that one Simpsons episode) so prosecution is unheard of. At least, until we met our current surveillance state dystopia

Also in NY the money is not inherently turned back to the finder. Found money goes into the city treasury unless the finder stakes a claim. But this would likely take so much time and paperwork headache that any feasible currency you would find would almost certainly not be worth the effort (maybe if you find some mafia dummy’s billfold, I guess, but then it’s approaching “wallet” territory anyway and also probably a bad idea to keep for other reasons)