No, the coin you have in an exchange is theirs. You don't have those private keys. And the coin can move around. Coinbase actually flaunts this as a feature

No, the coin you have in an exchange is theirs. You don't have those private keys. And the coin can move around. Coinbase actually flaunts this as a feature (you can give money to other Coinbase users without using the blockchain). This is also why they won't accept generated coins from mining payouts.