Yes, much like admin.google.com (the GSuite admin interface), which goes ahead and tries to two-factor your personal GMail account every single time you load it instead of asking you which of the actual GSuite accounts you're signed into you'd like to use...
Yeah, with multiple chrome profiles, you have to be mindful of which one you last had focused before clicking a link from an external application (i.e. tailscale), so that it opens the new tab in the right instance so the account(s) you use in it are available
Def use multiple chrome profiles if you aren't. You can color code them to make visual identification a breeze
I always have a buggy ass hell experience with having multiple google accounts pretty much across all their services. I've been wondering if its just me or how the hell this is normal.
I long ago concluded that trying to mix multiple google (or MS) accounts in the same browser profile is a path to madness.
I love how the two factor screen has no obvious way to tell it that you want a different account.
Hint: you can often avoid some of this mess by adding the authuser=user@domain to the URL.