I will shock you at first. No one crypto wallet stores your funds.
The wallet is a software or hardware, which can connect to blockchain and gather your money accordingly your private keys. As far as you have your private keys - you are safe. You can have them on paper, stored somewhere in grandma’s chest.
ERC20 is a protocol on Ethereum network to allow you to have your own assets, not related to the underlaying network. They can have individual pricing and usage. But to move them through Ethereum network from the one address to another, you need to have ETH to pay fee to Ethereum miners.
This protocol is designed the way that any wallet which is compatible with ERC20 tokens can see all of them or have an ability to register a new one if it’s not recognized automatically.
However, if you use a deposit address of any exchange you do not have private keys. This is mean:
- money on deposit is not yours, they belongs to the owner of private keys (Coinbase in your case). And the owner will decide - would it allow you to access them or not.
- most of exchanges uses a separate address for every supported coin/token, even if the ERC20 tokens can be send to any Ethereum address. This is related to how they work with deposits. As soon as your money arrives to the deposit, the special dumb robot which can recognize only one token will take them from there and move them to the exchange’s main wallet, updating your balance with the same amount. The balance is completely virtual. They do not owe you anything - you are give them your money in your good will. So, they can do with these money what they want. Maybe even allow to withdraw them. Sometimes. And with cut.
- since the deposit address is disposable, they can change it without any notification. If you sent money to a previous address (or to the address for different coin/token), they would be more like lost. You can file a support ticket to the exchange and if they would be mercy they can recover them. Or not.
So, in most cases it’s much better to have your own keys from the address, this is guarantee that while the blockchain is exist, you are the owner of funds. Wallets are disposable too.