It’s not so clear. I know of two men (Ashkenazi) with the last name Cohen who are not Cohanim (they are Yisroelim). I know of a fellow whose last name is Levy and he is a Cohen. I also know of someone who has last name is Levy and he is a Yisroel.
As for Shapiro; my understanding is that it comes from Speyer; one of the original 3 cities along the Rhineland in Germany during the time of Rash”i and Tosafot. That might explain why there are Shapiros who are Cohanim and Yisroelim. (I haven’t met Leviim with that last name).
Among Moroccan Jews, you can be fairly certain that the last name Cohen is a Cohen and that Levy is Levy (personal experience). What I found interesting is that the last name Serfati/Sarfati/Serfaty, etc… is prominent (among Moroccans). Serfati means French (or Frenchmen) and in some sources was the last name adopted by Rabbenu Tam. Yet everyone I spoke with doesn’t seem to see any Yichus to Rabbenu Tam (and by extension to Rash”i).