Request from a user: It would also be great if we don’t have to sign in on the extension on every tab.
You shouldn't need to. This sounds like a #bug. Let's see if we can diagnose what's happening. Could you check to see if cookies are enabled on your device?
With cookies enabled, you only have to sign in once per browser while without cookies, you would need to sign into each extension every time you open a new tab.
There are various reasons why Chrome might sign a user out. Here is a website that list the common reasons: https://techpp.com/2022/08/27/how-to-fix-chrome-keeps-signing-me-out-issue/
Note that each Method in the article is for one potential root cause.
Like Parker and Erika mentioned, the most likely reason is that cookie is not enabled, which corresponds to method 3 in the article.
I agree with the theory above - it sounds likely that cookies are disabled in this case. Please make sure you are on the most recent version of the Chrome Extension. Versions >= v0.0.12 should alert you if cookies are disabled.