Reviewing Manuscript Which Misinterprets My Own Work

Give the corrections, referring to yourself in the third person where needed. You don't need to impersonate someone uninvolved, you just need to depersonalize your corrections. For example,

In the introduction, the author is speaking of bla-bla; s/he motivates bla-bla by citing such-and-so.

Make sure you focus your corrections on errors that are relevant to the paper you are reviewing. That may mean that your list of corrections will not be a complete one.

For all the author of the paper you're reviewing knows, you might be an advisor or close collaborator of the author of the paper that was mis-cited. That's okay.

Note, if the author of the paper you're reviewing forms a conjecture that the reviewer is actually the author of the mis-cited paper... that's okay. Maybe someday you'll meet each other at a conference and enjoy a moment of mutual recognition.

I am basing this advice on my own experience writing reams of legal documents about a family member, whom I got in the habit of not naming in my writing, to protect his privacy. I also wrote about myself in the third person in those documents. I found out that with a little practice it becomes second nature.

It is very often the case that it is quite obvious (from his comments) to the author who a referee is. Many special subjects have just a handful of well-known experts in the world. You probably know them personally even, their educational background, the typical tilt in their English stemming from their mother tongue, etc.

With many journals, you propose potential referees, and the authors of that paper probably proposed you.

So: Don't worry about them deducting your identity. They won't mention it, and they likely won't be surprised. Pleased probably, to have gotten first-hand feedback.