Does an ER bridge between two black holes mean they are "entangled" and what happens to the ER bridge if a merger happened?
The Einstein-Rosen bridge is a property of the maximal extension of the Schwarzschild geometry. It doesn't exist for astrophysical black holes that form by gravitational collapse.
I wondered in the case that these two black holes were already entangled
"Entangled" isn't the right word here. Entanglement is a quantum-mechanical thing.
If you want to know what happens when the two event horizons in an Einstein-Rosen bridge collide, then the answer is that they can't collide. One is a past event horizon and one is a future event horizon.
If you want to know what happens when the two mouths of a traversible wormhole collide, that would be a different question. To make it a meaningful question, you would have to describe what wormhole solution you have in mind.
Yes, theoretically, an ER bridge between two black holes means they MIGHT be entangled. There is a conjecture called the EPR=ER Conjecture. The original paper, called Cool Horizons for Entangled Black Holes can be found here:
by Leonard Susskind and Juan Maldecena from 2013.
However, I want to caution that this conjecture has not been empirically tested yet, though it is theoretically testable in quantum computer networks, as it has been suggested by Leonard Susskind and Ying Zhao here:
If the tests regarding the EPR=ER conjecture yield validating results, then yes, provisionally, two black holes, connected by an ER bridge, are in fact entangled.
As for what happens during a black hole merger between the two black holes to the entanglement or ER bridge, I don't know at this time. I would recommend reading the original paper listed, and try to derive the answer from the behavior of the black holes in the conjecture. I have my suspicions as to what might happen, but will not include them in this answer as they are only opinions at this time.