Is asking an author I'm reviewing to cite me a conflict of interest?

If I were you, I would ask myself the following question:

is it better for the paper to be aware of my work?

I believe this is the main question you need to ask. If you see your work is relevant to the problem and can enhance the paper content, then you have to point to it. If you see your work as a complement of the paper just reference it in the review.

I don't agree that this is a conflict of interest. The reviewer can point out related work, and recommend that the author cite it. if the work is in fact irrelevant, the author can point this out, and the editor can adjudicate. Referees are supposed to be experts on the material.

This scenario has happened to me a few times. I've recommended that the author cite the relevant papers (including my own as needed).

Yes, of course there is a conflict of interest in this situation. Reasonable people may differ in their opinions on how the conflict of interest should be addressed in this circumstance, but I think it's obvious that a conflict exists. You are in a position of power over the authors of the paper in your role as a reviewer. If the authors were to cite your paper, that would be of some benefit to you. Therefore, asking them to do it inherently involves some element of conflict of interest.