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A wallet is two things:

- a private key or set of private keys which are used to sign transactions - hardware wallets store this inside the "hardware wallet" thing

- a program that interacts with the Bitcoin/Ethereum network, can display balances on a (public) key or make transactions that need to be signed by a private key (those are different, you'd use e.g. Electrum for Bitcoin and Metamask for Ethereum)

Obviously both of these can be the same (you can store your private key in a file accessed by Electrum), but to reduce the probability of getting hackes people often separate the private keys from their wallet programs.

A coin-denominated account on an exchange is often also called a wallet, but it's just an account on the exchange like you'd have one in your bank.