Time Machine issues when disconnecting Studio Display from Mac Studio

I got a Mac Studio a year ago, along with an Apple Studio Display, for my freelance video work. Since then, I started a new remote job that requires me to use the employer's provided PC laptop for privacy reasons.

Rather than using the low-quality monitor my employer provided, I am using the Studio Display with my Mac for freelance stuff, but then for the new job, I unplug the cable, and plug in my PC work laptop to the Studio Display. So I'm using the same Studio Display monitor for both computers at different times.

Somewhere during that switch back and forth, it seems that my Time Machine backup drive stops doing the automatic backup on my Mac Studio. Does that make sense? Is there a setting I should change?