Updated 2020-11-27
Happy to see that SourceTree allow users to skip the login procedure in version 3.3.9.
For older versions:
SouceTree uses accounts.json file to store account settings.
It would skip the login process if there is an accounts.json file in your machine.
So you can copy the accounts.json file in %AppData%\Atlassian\SourceTree\ to any other machine to skip the login procedure.