ERC20 is different from Ethereum, the ERC20 wallet supports all coins in the ERC20 shelter. I suggest using MEW (myetherwallet.com) it is the best wallet in my opinion.
ERC-20 is a type of token on Ethereum blockchain, there is no much difference, and basically all ETH addresses can have ERC-20 tokens on them, but if you do not have private key (like you do not have for exchange wallet), you cannot use them for your own purpose, so they will be lost for you
you can check when you make a deposit on exchange, it is the same address for ETH and all ERC-20 tokens that you can deposit on your exchange, but if exchange do not support some tokens, they will be "out-of-reach" for you with exchange address