Does anyone know what part this is?

Depending on the series you’ll encounter a few different setups.

This is both an EMI and electrical shield between the display and the S module which is a bit noisy. The bracket is the chassis ground point the sheet is clamped by it.

Apple never offered watch parts externally and most broken watches are shipped to China to be refurbished. So when you drop off your watch off at Apple they just hand you a refurbished watch. Sadly, this series is no longer repaired. Apple will offer you a dollar amount to put to a new watch.

You might find another watch to recover the part. But that doesn’t explain why this watch is not working. Something else is going on which needs to be figured out.