I’ve been stopped with no “real” ID and the police get really thorough all of a sudden. Even though I could provide a name, driver’s license number, and had a debit card on me, they run your name and then ask you about any names that are anywhere close that they might be looking for. “Do you ever go by Randy?”... no. “How about Sam?”... no.
I believe just the act of not identifying yourself is enough to get you arrested, and then I would imagine they fingerprint you and try whatever they can to identify you. You can be arraigned without identification, and I would imagine a judge would compel you to identify yourself or stay in jail.
Since they already have you physically, I’m not sure if there’s any hurry after that.
Edit to add – stop and identify laws which seem to vary by state.