: regulate the stay and regulate the ability to work.

Permits don't regulate anything- They are legal proof that you are allowed to stay and work. Laws regulate things.

It has much to do with the residence permit and OP even quoted that part in the question. Persons with a student residence permit are not allowed to work more even if they wanted to under any circumstances.

I don't know what you are reading, that's exactly what I wrote. Of course they can request to work more than 120 / 240 half-days. If this request is accepted is a different story, but it's not prohibited by the law.

but also work illegally under residence law.

Yes that's what the resident permit and work permit says "allowed to stay for a year to study and be able to work 120 days. That's similar to the restrictions German students have, although maybe stated in a different law. So there is no difference between foreign and native students. All are treated the same. Though, of course if a foreigner doesn't study, the visa would be invalid and therefore the limited-hours work permit too.

https://www.gesetze-im-internet.de/aufenthg_2004/__16.html

There is no salary ceiling. Maybe it arouses suspicion but in the 20 hours you can earn as much as you want (or rather can).

If you receive additional funds (Bafög), there is a limit. That's why I wrote at the end, it may bot be a problem for a master student with a job, but for him it probably is.

As a foreign student you have to report the Ausländerbehörde if you intend to work more than your current work permit allows

That's basically what I wrote just a few words later.