PHPSESSID is a constant that differs for each browser that loads a script that uses sessions. As such, each session is assigned an identifier (read "big long alphanumeric string") that is unique to that browser.
Note that the manual's prototype uses a "void" argument. You will get some type of error if you send a "name" to [man]session_destroy/man.
It may not seem clear, but it's because the concept of the session is likely not fully clear to you either. As I said above, each session is unique, and session_destroy() will only clear the data from the session that it is currently involved in; unless, in the case of assigned session_name(s), you assign the same session to more than one browser by accident. (I.E. if you call them all "MySession" (string) by accident, for example...)
Otherwise, there is no danger to other browsers that are on the same site, even the same script, at the same time.
HTH,