They are a subset of mathematics and should be treated as such.
In other words, the answer to "are functional equation problems on-topic" is the same as the answer to "are maths problems on-topic": namely, it depends. We have an established policy on which maths puzzles are on-topic and which are not:
So, what makes something a math puzzle rather than math problem? I think there's a few features.
- Clever or elegant solution, often an "aha" moment
- Unexpected problem statement.
- Unexpected or counterintuitive result.
In contrast, math problems tend to be "textbook". And by that I don't mean that they have to come from textbooks (or that textbooks can't contain math puzzles), but that they use standard, staightforward methods than anyone familiar with the subject is expected to know. They can be difficult, but their goal is to test comprehension of the material, not ingenuity. This doesn't apply to problems from math olympiads like the Putnam exam, which are designed to have clever solution.
Most functional equation puzzles are on-topic.
Of course there are examples of trivial functional-equation problems which could be closed: solving something like $\frac{d}{dx}f(x)=f(x)$ (yes, differential equations are functional equations too) is a pretty standard problem and not really a puzzle.
But Olympiad-style functional equations, like most/all of the ones we've seen here so far, are likely to be counted as maths puzzles, like most other Olympiad-style maths questions: thinking outside the box, not using standard straightforward methods, is rather the point.
