For example, does the Engine hand out a new connection from the Pool on every query?
only if you have autocommit=True, which should not be the case.
Or is it attached to the session for its lifetime, i.e. until the request has been processed and the session & transaction are closed/committed?
it's attached per transaction. But the "search_path" in Postgresql is per postgresql session (not to be confused with SQLAlchemy session) - its basically the lifespan of the connection itself.
The session (and the engine, and the pool) these days has a ton of event hooks you can grab onto in order to set up state like this. If you want to stick with the Session you can try after_begin.