Your subscription license is tied with the google account that you purchased the license with.
For the extension to be activated with your license, the Chrome browser needs to be signed in with the same google account.
When you sign your Chrome out and disconnect it from google account, or sign in with a google account, there may be a risk of loosing all your extension data.
In order not to lose your data, there are 2 things that you should be careful with.
In order to purchase Force.com LOGINS extension license, you need to have valid google account with google payment service enabled.
When you click on the "Purchase license ..." button in the extension's option page, it tries to open google payment window with the google account your Chrome (not browser tab but Chrome browser itself) is signed in with.
You can check your Chrome's sign in status at chrome://settings/.
Apparently Google Chrome sometimes mess up the google account that Chrome itself has been signed in at chrome://settings/, and the accounts you signed in with the browser tab.
That could give you an error when you try to purchase the extension license.
This article talks about some simple steps to straighten it up.