yes, the mint fee is to compensate for the work it does.

If two people are paying each other across mints, then they'll pay for the mint fees + lightning fees

To understand why let's look at the steps involved in a normal transaction.

1) Alice has an ecash banknote on MintA

2) Alice sends it Bob

At this point, both Alice and Bob own the ecash banknote. This is problematic, because Bob wants to have unilateral control of that banknote.

To do that Bob

3) Request a new banknote from MintA. MintA invalidate the old banknote, and gives a new one to Bob, that only he owns.

So to prevent MintA being DDoS attacked, MintA charges a fee for every request.