If I play Abandon in response to a card being played from discard through the flow mechanic, does it go back to trash or does it become banished?

It becomes banished — it will not go to the trash, and it will not even go back to hand despite Abandon's text.

Abandon's "return to hand instead of trash" and Flow's "banish it instead" are both replacement effects modifying the same event (the spell leaving the chain), so the spell's owner picks the order (Rule 372) — but either order ends in banishment. If Abandon applies first, the spell would go to hand, but that still counts as "leaving the chain," so Flow's delayed replacement kicks in and swaps the destination to banishment. FAQ #11690 confirms a countered Flow spell is banished, since Flow's effect covers any departure from the chain except the spell's own execution (Rule 829.1.b.1).