> Is there nothing like some sort of number to register any tax withholding or the like?

There is, but it's not tied to any strong identity verification process, and so there's a thriving fraud where unemployed citizens will rent out their numbers to working illegals. It's not something that the tax office has ever really worried about, since if anything it tends to increase the amount of tax paid (if several people are sharing the same tax ID they'll pay a higher tax rate), and while they might bat an eye at someone with 5 different salaried jobs it's not particularly suspicious when it's gig economy work.