In a private wallet like Exodus or a hardware wallet like Trezor, you can only hold the tokens that they support. ie. if you want to put Illuvium tokens there, and they don't support that token, then you will not be able to move your Illuvium tokens from the centralized exchange to your private wallet.

Regarding Metamask, some say it only support Ethereum based tokens ? I'm keen to use MetaMask, but I've also heard horror stories about YouTubers who got hacked using MetaMask, one of them lost almost everything ! ( I think they were staking ) So with MetaMask the risk of getting hacked seems too high !