Does Wittgenstein's philosophy have metaphysical assumptions?

I think it does. When arguing for a form of "cluster descriptivism" in the Investigations (sect. 40), Wittgenstein implicitly assumes an A-theoretic or presentist view of time. An eternalist need not hold, if they hold that the meaning of “N.N.” is N.N. himself, that the meaning “dies”, in the sense of going out of existence, when Mr. N.N. dies. For according to the eternalist it is never false that N.N. exists, only that he exists now. So the term “N.N.” will refer to something past (relative to our current temporal stage), much as the phrase “the battle of Waterloo” does. So someone who believes proper names directly refer can remain consistent if they also embrace eternalism.

Wittgenstein's criterion of understanding as a person going on in a certain way presupposes some notion of transtemporal identity. It will not do for Wittgenstein to say, for example, that Bill’s understanding of an expression e depends on the subsequent use that Sally makes of the expression e. We can usually tell people apart, but certain neurophysiological conditions, e.g., split brain cases, throw our ordinary concept of personal identity into question. And even aside from neuropsychology, difficulties concerning persistence, such as the (in)famous Ship of Theseus, make the case for perdurantism over endurantism. Given the truth of perdurantism, as well as the mereological complexity (non-atomicity) of persons, it is to some extent arbitrary (or at least vague) which momentary entities are con-perdurants with which. Wittgenstein also cannot account for a person’s understanding or their “use” of a term in an antirealist or conceptualist manner by invoking our understanding or use of such terms as “identity”, “person”, or “same”, on pain of circularity. So a Wittgensteinian must give us some non-circular account of the perdurance of persons before they can explain a certain person’s understanding of an expression by how they, and not someone else, go on to use it. And if the notion of numerical identity is senseless, as Wittgenstein seems to suppose, then Wittgenstein’s account of understanding is senseless if he in any way relies on it for his account of understanding, as it would if temporal identity is ordinary logical identity. His account seems be in hot water irrespective of whether transtemporal identity is numerical identity or not. It's stuck between the rock of vicious circularity and the hard place of self-contradiction.