I was also part of this sort of interview once. They specifically asked personal questions - parents stuff, relationship, etc. Definitely not work related. It was indeed a very strange and exhausting experience. I could've definetly refused to answer some of the questions or drop out of the interview altogether, but not sure why I haven't.
So yeah, this type of interview exists so I highly doubt the interviewer interviewing OP was asking about work stuff...
I've built this kind of extension, but the backend being Linear (the issue tracker). I am able to mark an email as an issue, start it, complete it and see details about it, all within Gmail and synced with Linear.
I guess using a local logic should also work - basically having everything inside the browser / extension.
So yeah, this type of interview exists so I highly doubt the interviewer interviewing OP was asking about work stuff...