Hi ahilmort,
There must be something more than that going on... because pressing "Backup Now" sends zero information from the browser to the website, except the information that a backup should be scheduled in 5 seconds time. You can verify that by using something like WireShark to monitor your network traffic if you like. Information about what Google account your browser is logged into is never communicated to the webserver either positively or negatively, so it's quite impossible for the webserver to do anything either way in response to it. The server side neither knows nor cares what your browser is doing.
Secondly, though, I can't reproduce that. I did this:
1) Logged into Google using one of my Google accounts
2) Opened up the UpdraftPlus settings page in another tab in the same browser.
3) Press "Backup Now" and waited for the backup to complete, which it did.
4) Verified that no backup was in the Google Drive for the Google account I was logged in to.
5) Logged out of my Google account, and logged into the other one, and verified that the backup was there.
So, some other subtlety must be involved in your case.
David