No, you hold the eCash. The mint is simply the issuer. It works a lot like the way USD used to back when banks could print their own reserve notes. You hold the eCash tokens yourself as a bearer instrument, and the mint holds the underlying sats.

As long as the total amount of sats they hold and total number of tokens issued matches, they are solvent. If they issue more tokens than sats... Well, someone bout to get rugged.