Do professors get paid for supervising students?

In the US, supervising graduate students is generally considered part of the normal workload of a faculty member and there's no extra pay for doing this.

The number of students supervised is typically a factor in tenure, promotion, and pay raise evaluations. Not supervising enough graduate students can hurt your evaluations and might possibly result in your not getting tenure or promotion or a pay raise.

Not in the UK. Supervision is usually a job requirement. It's also necessary to boost research output.

Although professors are not technically paid for this work in the US, in many cases they are effectively paid for doing so, particularly for Ph.D. students. This is because in many cases, the professor requires grants in order to be able to hire Ph.D. students to work for them, and those same grants pay for a portion of the professor's time, some of which is expected to be used for supervising the student. The accounting is often rather obscure, however...