I'll just have to guess that most places will have regulations concerning, or even forbidding, this. But it is a big and variable world.
But, at a minimum, a person contemplating doing this needs to check with the university administration, who may permit it or not. Or they might set some boundaries. I doubt that many would permit tutoring students at the same institution for pay. They would rather provide tutoring opportunities for grad students, for example.
I know of one case in which a person was fired for doing such things, but mostly, I think, for not revealing the relationship, which was with another school.
But if you can justify that the tutoring is completely independent of your university duties and doesn't compete or interfere with it, then you probably can make a case that it is OK. The university doesn't own all of your time and what you do with it is a personal matter - up to a point.
I'll also note that there are other educational activities that are somewhat similar and most universities don't interfere with them. Writing textbooks, for example can make quite a profit for a few professors and I don't know of universities that get in the way of this. Some activities are even encouraged, such as consulting to organizations or businesses.
But note that the activities that are encouraged tend to have a repetitional benefit to the universities. Tutoring for pay doesn't seem to fall into a category like that.
Adjunct professors, of course, normally have to teach at more than one institution, provided that they want to eat and pay rent.
Also, being a professor (at any level) is normally more than a full time job. It is difficult to do everything required and have enough time for much else, especially if that other commitment requires a schedule.