Syntactically, they're both determiners. Semantically, they're a bit different since possessives (like 'my') aren't closed under permutation†, while many other determiners ('some', 'every', 'most' &c.) are closed under permutation.

† An operation mapping a set X onto another X' by relating every member of X with some member of X', so that X' has the same number of members as X, but the identity of the members may be different.