So each coin already has a serial number?
No and yes. There is no serial number. There is
1. The bitcoin address -- that you send BTC to load it up and
2. It has a private key that you use to spend the money associated with the bitcoin address.
This is something I also struggled with at the beginning: don't think about bitcoins like coins! Let me try to explain it in an easy way:
There are no "coins" but "balances". The pubkey (and the correspondent private key) doesn't identify a "coin", but your wallet. And this wallet has a balance (2,7242482 BTC for example). When you transfer some BTC to someone, you transfer your balance (or part of it).
Does it make sense to you?
