Does Named Credentials provide a way to store returned access tokens?

When using OAuth, the Named Credentials automatically stores the Access Token and Refresh Token. When making a callout using Named Credentials, Apex will automatically try the Access Token first, and if it gets a 401, with an error response of invalid_token, it will try the Refresh Token to get a new Access Token, and try again, and if that fails, then throws the appropriate error. This all happens behind the scenes without any effort on part of the developer.

For non OAuth tokens, such as API keys or something else, you need to arrange for your own storage. This could be a custom setting, custom metadata, platform cache, or another solution. You'll need to remember to keep this information secure by access controls and/or encrypting the token.