Is private key the same as the 12 recovery phrase?

[MarkRoberts]

It is not the same exactly. The seed is used to generate the private key. You don't actually interact with the private key directly, but the seed is all you need to recover the private key, so yes.

I don't have experience with Exodus personally, but assuming they are not just scamming and giving you a fake recovery seed, you should be fine. For a list of wallets which allow you to be in charge of your keys, google 'non custodial wallets.'