I think many DON'T pay their taxes. If you own a house and don't pay your taxes, the IRS can try to take it for payment.
I also see your play on words with how does the bank "know" they are illegal? Well, if they are applying for a loan through this program for illegals, it is a good bet they are illegal. Also if they are using a TPID number, it is likely they aren't a citizen and possibly illegal.
Banks dig deeper into people's past and background when making loans, particularly loans for homes of the price bracket in San Diego. Employers don't have these resources and it doesn't really make sense to do so for a minimum wage employee. Would you apply for a loan if the application process cost you more than several months payments? Banks know more about their applicants than most employers know about their employees.
I doubt many illegal immigrants make enough to owe ANY federal taxes.
As for "Would I apply for a loan if the application process cost you more than several months payments?" I don't know what the exact cost, points, etc. was on our recent Countrywide loan (and no, I am not an illegal immigrant) but we did end up paying quite a bit to get the lowest interest rate possible.