The only way to log into an existing MetaMask account on a different device is by using a Seed Phrase. You can find the seed phrase or the secret phrase in Metamask>settings>Security & Privacy>Reveal Secret Recovery Phrase> Enter Password. Once you have it with you, you can follow this tutorial to log into your account on any device.
To import your account in MetaMask:
Ensure that no account is logged in on your device.
Find the extension icon on your browser and click on that.
Locate the Metamask extension and click on the three vertical dots near that.
Click on the Remove from chrome/(your browser) option to completely sign out of Metamask from your browser.
Install the extension again on your browser. You can follow these(hyperlink) steps to know how to install the extension. After a successful installation, it will greet you with a new window. Click on the get started button to initiate the process to import the MetaMask wallet.
MetaMask will ask you if you want to share some of your usage data to improve its service. If you are not sure what to do here, just press no thanks and proceed to the next steps.
The next screen asks whether you want to create a new MetaMask account or if you have an account that you want to import into an existing wallet. We will smash the “Import wallet” button and keep our seed phrase ready for the next step.
Fill out the “Secret recovery phrase” or the “Seed phrase” of the account you want to import in the correct order. Enter and confirm the new password for this account and click the import button.
Congratulations, you have successfully imported your MetaMask wallet. Click on the “All done” button on the next window and start using MetaMask.