This worked for me on macOS (... only)
Go to Repository > Repository Settings > Remotes and edit the repository path by adding the user name (like https://<user name>@github.com, by exemple https://trentreznor@github.com/landron/Problems.gitinstead of just https://github.com/landron/Problems.git). Then, at the first push (tentative), SourceTree will ask for password (and store it to Keychain if you're on macOS). I found the idea on the Atlassian support site.
Windows update I did not succeed in doing the same because I did not convince SourceTree to make/use two accounts in Credentials Manager: git:https://<user1>@github.com and git:https://<user2>@github.com (they can be created manually by "Add a generic credential" in Control Panel\User Accounts\Credential Manager).
Identification update @Simeon Visser's answer matters (at least) for github statistics (contributions) because github identifies the user by email address (see "The email address used for the commits is associated with your GitHub account." here).