Can university educators provide private tutoring to anyone seeking it? Let me elaborate: I’m aware that school teachers often do private tutoring for children on the side (although I believe it’s only ethical/fair if they tutor children from other grades, classes or different schools).

Are university professors (at any level) able to engage in a similar job? Could they offer private tutoring as a paid service to students from different classes or institutions, or even to children and adults in general who need tutoring? Would this breach any university policies or regulations?
Many university professors have side gigs doing consulting work for industry, writing, starting companies and doing many other things. At all universities I’m familiar with this is permitted and to some extent even encouraged. Usually there will be a policy in place specifying how much outside work is permitted. For example, here is a link to the relevant policy at the University of California system.

As for your specific question, tutoring students at other universities is not any different from any other kind of side work. As long as the professor is in compliance with their institutional conflict of commitment policy, this is totally fine. The only other case where it would be a problem is if they want to make money from tutoring students at their own institution.  This would create a fairly blatant conflict of interest and would likely be regarded as unethical. See for example this question.