Yes, the mnemonic is the only thing that defines the private key of your wallet. If you somehow will get to know someone's mnemonic phrase, you will have full access to their wallet. That's why it's so important to keep the phrase secret.

I should also note that it's possible to generate a private key without using a mnemonic phrase. Commonly used wallets usually do not expose this functionality (because you won't be able to back up your wallet by writing that phrase somewhere), but theore...