I'm having the same problem. Pixel 4a with Android 11, HP Spectre x360 with Windows 10. All software is up to date.

There's two possible improvements that would make it much easier to use Android Messages on the Sidebar:

1. Implementing Android Messages as a native option for messaging, alongside Whatsapp, Instagram, Facebook, etc.
2. If that's difficult, then allowing custom websites added to the sidebar to send pop-up notifications. (An added bonus would be enabling shortcuts to be set for particular sidebar extensions.)

As it stands, Android users have two options for messaging from PC. The Phone Link app lacks encryption and hi-res media, so that's not great. Using messages from Web requires an extra tab to be open, or an entirely separate window (if installed as a Chrome/Edge PWA). The tab isn't ideal because it gets buried. The PWA isn't great either because links end up opening in which ever browser set up the PWA, even if Opera is the default browser.
Using either of the suggested implementations would allow Messages to be running while Opera is open, with notifications, without the need for a different browser.

@petergroft I didn't realize cookies were disabled for those sites. Turned them back on and can now log in. Thanks!!

Is there a way to prevent the cookies from being used for tracking, though?

@leocg I get error messages from the page I'm trying to log in to, to the effect of "this request couldn't be processed" or simply "oops! something went wrong" (with variations for each website, of course)

@leocg Yep! I gave it a shot after disabling all the extensions and resetting the whole browser. That just made it worse because then I got logged out of everything, not just the first couple websites. And now I can't log back in.

I’m getting error messages when trying to log in to certain websites. I first noticed the issue on YouTube, and tried resetting the browser. I still can’t log in to YouTube, and now I can’t log in to Facebook either. Both of these sites still work with other browser or incognito windows.

I'm having the same problem. Pixel 4a with Android 11, HP Spectre x360 with Windows 10. All software is up to date.