You can get a tax identification number without an SSN.

An illegal / undocumented worker working a standard W-2 paycheck job is going to have taxes withheld and sent in by the employer, even if they never file their own tax return.

It is not like that, usually immigrants live in grey arey keeping all contacts with govt to the minimum, the whole idea of hiring illegals is to avoid taxes. I know that from personal experience of being illegal immigrant in US of very close person.