Your coins are not saved in the wallet; they are being saved in the blockchain and not the wallet. That is why you can use your generated private key to assess your asset in any wallet that support the private key, and these wallets are safer than leaving your coins in an exchange where you don't have control over your assets.
Storing bitcoins on centralized exchanges is one of most risky practice, because you can lose your bitcoin. The exchange owns your bitcoin, and they secure it for you or steal it from you, you can not control their activities.
Reminder: do not keep your money in online accountsAlways remember that not your keys, not your coins.
It's correct.
Bitcoin Q&A: Not your Keys, Not your Coins.