Like quid, I'm a regular reviewer, with about 60 reviews to date. The large majority of those reviews have been untouched since they first appeared, but in a few cases I have later asked Math Reviews to implement light edits to correct errors. The edited reviews appeared in due course. (That is, they appeared in MathSciNet; I don't know the policy about the print version.) On one occasion, I did this because the author contacted me to point out a mistake in my review, which I was glad to correct.
In writing a review, I'm aiming for concision. But I try to touch on 1) the context of the work, 2) the main results, and 3) what sorts of ideas go into the proofs. I read only in as much detail as I need to do that. Occasionally, when I feel that I have a distinctive perspective on material in the paper I'm reviewing, or its relation to other papers, I comment on that perspective. I try to do so sensitively and sparingly: I am conscious that these comments should add to the information I convey about the article, not replace it. But I would argue that as a reviewer it is my privilege to explain the context of the work as I see it.
If the problem could be resolved by straightforward corrections to the review, I would suggest that you contact the reviewer directly. If the review is so misleading as to make you think that the reviewer shouldn't be writing about papers in that area, you could instead make that case to Math Reviews - and perhaps enlist your expert colleague to do the same.