If the airdrop project requires you to use certain wallets, make sure that you know what you're downloading. Because many of these airdrops will want you to download wallet that has malware in it.
But if it's only asking you for a specific wallet address then all you have to do is to provide the wallet address.
You have to be more careful in this case, because a new, unknown wallet will be potentially dangerous, unless it only asks for an EVM ( Ethereum Virtual Machine ) address or an address for a specific network according to the project being worked on.
But when the OP says about using an Exchange address, it's not recommended even if the wallet needed is a SOL wallet, or even an ETH wallet.
Since the Project is not listed on the Exchange, the coin will not be transferable.
Many of these projects will take advantage of their participants and that's why they have to be careful with that. Otherwise, they're putting themselves in danger and much worse, if they have deposited any amount on that unknown wallet, they'd lose that money.
Use a private wallet, a wallet that is created by yourself and has a private key.
The term is not your key, not your coin.
Simple as it is and the tip never gets old.
I hope that most airdrop joiners will have to remember this and for the likes of OP.