I recently received an invitation to review a paper for publication in an MDPI journal. This is the first time I am being invited to review. I do feel qualified to do it since I have several publications relevant to this topic. However, in the invitation e-mail, I am greeted as Dr.- which reveals to me they are under the impression I hold a PhD, when in fact I am still an undergraduate. Should I decline and correct them or should I accept the invitation? My advisor feels it would be good for my CV but I have qualms about capitalising on a misunderstanding.
You have several publication related to the topic. You were selected as potential referee for your work not for your title.
One can eventually discuss the correctness of the journal (though, why the journal should know if your contributions were made by a student or a doctor?) but you can certainly accept the task if you feel that you can review the paper. (Note that this shall be done by every referee no matter of experience).
Your supervisor is right. Eventually, you can ask her/him if you feel too soft or too harsh on some points, or whenever you feel unsure. A common behaviour of young scientists is to overestimate minor details, or asking for citation of whatever contribution they have made so far, etc. The supervisor can likely find some minutes to go through your report.
Often, while taking the final responsibility of the report, supervisors assign part of the review of proceedings or papers to suitable students in their group as a way to make them familiar with an academic research aspect. This time, you were directly invited. So have a look at the manuscript. If you have something reasonable to say, go ahead.
Edit:Thanks to the comment by Wettenschaap I am not so sure my answer still applies. While I can repeat every single word as far your concern (being just a student) is considered, I am not sure if it is worth putting time and energies in reviewing for a journal of that publisher. However, you won't be oblige to put your referee activity in a CV, or you can mention it without giving much details. So you can still do the task and get the related experience. If so, do it seriously.
I am greeted as Dr.- which reveals to me they are under the impression I hold a PhD, when in fact I am still an undergraduate
What probably happened is that the editor(s) searched the literature for related papers, and then assumed all the authors have PhD degrees. After all that's most probable title for the authors. "Prof" is the other title, but there are a lot of authors who are not professors. They could have researched you in more detail, but with a hundred reviewer every day and not every author being easily Googleable (in my experience, this is especially the case for Chinese authors) this assumption saves a lot of time.
I'm with your advisor here. If you are qualified to review, you should, and you can put it on your CV. Wetenschaap mentioned in a comment that you might not want to put anything MDPI-related in your CV, but it really shouldn't cause much harm if at all, because you are doing reviewing which everyone agrees is something that needs to be done.