You have a few options.
If you ONLY need some user id then Google's OAuth2 API allows you to do so. This user id is unique for users, and you can use it (for instance) to check if a user logged on before. Or for communicating with Google.
If you need more, like the actual email address, then you also have an option: When the user is about to log on, you have to pass a so-called scope to the OAuth2 API. You can actually pass multiple scopes, and the scope for the gmail API can be among them. If you ask two scopes, and the user logs on and agrees, you get a single access token allowing you to access the account with both APIs.
So what you do, you pass the scopes for AdWords API and gmail API, and you use the access token (and refresh token, if necessary) for both APIs.
It has some side effects:
1. Obviously, when a user logs on, Google will ask him whether your software is allowed to access his AdWords account AND access his gmail account. Note that gmail API has multiple scopes (only reading / also sending / etc.), you better choose the minimal scope that provides the information you need.
2. When a user logs on, and you store its refresh token (for instance, to do large batch operations at night), do realize that this refresh token will be withdrawn automatically when the user changes his password. And possibly other options too, like reconfiguring 2FA. That behavior is related to gmail API, refresh tokens without access to gmail survive a change of passwords. (And that is not exactly true, Analytics API had the same behavior).
3. I am not sure if it works on accounts that are not gmail accounts. I would think you could still do a limited set of actions, like asking its properties, but I am not sure.
I hope this helps.