Typically a wallet lets you hold, send and receive coins. For example Solar is one such pure wallet.
An exchange lets you exchange them for other currencies, buy crypto coins for fiat (usd, eur).
It seems Lobstr confused you, becuase it kinda combines all of this together.
Do exchanges have wallets as well?
Sort of. You have an account on an exchange, this account has an address, and you can send from and receive to that account. But you typically don't have keys to that "wallet", it's kinda like having the money in a bank. If the bank disappears, so does the money