So, Syrio is able to come out of shadows at the beginning of the epic phase, I got that part.

If you play multiple epic battles cards, does he have a chance to come out for each challenge, or only the first one? Are they treated as separate phases completely with their own chance to bring shadows cards out?

From the FAQ:

(3.34) Multiple Epic Phases
If multiple epic phases are created in a single
round, they are played in the order in which
they were created, after the dominance phase
and before the standing phase.

So multiple Epic events create multiple phase, which means each would have its own "beginning of phase" opportunity to bring cards out of Shadows.

Thanks, I was familiar with that line of the rules. I just wasn't sure if it allowed a shadows action for each one.