Reviewer signs name on review. Should the editor censor?

If the journal is structured with a blinded review process, as most are in my experience, I would censor the name as an editor.

Only if there is some sort of explicit journal policy allowing reviewers to unblind themselves would I consider not censoring the name.

So, I have heard of people not censoring when people do so. There have been at least some controversies in some fields where this has happened. See for example, https://www.quantamagazine.org/a-self-aware-fish-raises-doubts-about-a-cognitive-test-20181212/ . For that reason I would strongly recommend removing the signature. Anonymity is important, and I personally (and other people) have had bad experiences with referees who have deliberately unmasked themselves. A big part of the concern where a referee has deliberately unmasked themselves is that if they are prominent in the field there's a possible implied intimidation or threat of retaliation for results they don't like. Also, it is possible that the file you got was intended for the editor and wasn't actually intended to not be unmasked in the first place. But regardless, editors should do all they can to keep the referees anonymous.

There are a few different cases to consider. First of all, there is the question of whether a journal's policy even allows for signed reviews. I think that most journals do not have an official policy about this. However, if there is a strict prohibition against non-anonymous reviews, then the editor should remove the identifying information before sending the report on to the authors (and any other relevant parties, such as other referees who are working on the same paper).

In the more likely event that signed reviews are not outright forbidden, then editor should look at the additional question of whether the referee really intended to make their identity known. From the report alone, it may or may not be clear whether a referee is intentionally choosing to dispense with anonymity. If there is just a signature at the end of the report, the reviewer might have added it out of absentmindedness. If the situation is unclear, the editor should check back with the referee, to see whether they actually intended to include their name before passing that name on.

However, I have seen one review that concluded with:

I choose to sign this review.

[Referee's Name]

In that case, it was quite clear that the reviewer (who was both a very senior person and giving a positive report) was not worried about maintaining anonymity. In a clear-cut situation like this, a referee can simply send the authors the report without any additional concerns.