The hukou system is nothing like that. It's an internal migration thing that works like moving to another country, you can gain "citizenship" of a different area and live there with full benefits.

It was also designed for a clear purpose (slowing down migration into cities that weren't ready to accept it) and was pretty successful, it just outlived its usefulness and became corrupt. Not the same as something based on ancient class divisions that was probably never that useful in the first place.

People work illegally because getting a stamp legally is hard if you don't have money or contacts in the local administration. Same reason as illegal immigrants anywhere else. The central government has been pushing to raise registration quotas and make the process easier but it's clearly not enough.