No, Adobe does not provide anything OOB for such a situation.
AA most likely has its own ID for identifying visits, but this is really for its own internal use and would never be exposed in any reporting interfaces (API or otherwise).
So you will have to come up with your own ID and track that to an eVar.
You mentioned that a user may use multiple products within the same session but not necessarily the same visit. I assume that
- "session" refers to the user's logged in session, where he could login at Day 1 at 9:34am and the session lasts until he logs out (or his login times out) on Day 7 at 11:59pm.
- "visit" refers to AA's visit definition, i.e. continuous period of time until no hit is collected for more than 30 minutes.
If the above assumption is correct, then you will also need to consider what ID you want to track: a session ID or a visit ID. If you're setting a session ID, then given that an eVar has a few options for persistence (Visit, Hit, a success event, Never, etc), you'll also need to think about how to set your eVar with your session ID properly to ensure that you're tracking the session correctly.