If A belongs to set A and set B, does it imply that A is a subset of B?

No, at least not in general.

You are a member of the set of people, but you are not a set or therefore a subset.

Obviously one can construct a set to which both x and {x} belong, e.g. the set that contains {1, 2, 3, and { 2, 3}, this being the four-membered set {1, 2, 3, {2, 3}}. But short answer—no. members not subsets.