For anyone affected by log-in problems who wants to help track them down, I'm going to quote a list of things to try (and potentially report back):

Quoting T169261#3452695:

Authentication bugs are notorious for having similar symptoms with different causes, since the symptom is usually "I can't log in" or "I get logged out unexpectedly" and there are many things that can cause that, including actual MediaWiki bugs, browser bugs, mysteriously-corrupted cookies that never get enough information to be investigated, and user error such as blocking first- and/or third-party cookies from our sites. And then people tend to assume their problem is the same as someone else's.

Which is why we usually wind up asking people with authentication problems to try several things.

Please see and follow https://www.mediawiki.org/wiki/Manual:How_to_debug/Login_problems and report back here. Thanks!

Does it persist after clearing cookies for the wiki domain? When logging in in incognito mode? When logging in with a different kind of browser?

Yes. I even tried clearing the browser data on Chromium. I experience the same when logging in using Firefox. Incognito mode has no difference either.

Does it affect all user accounts, or just one?

It affects my another account too.

Does the "remember me" flag make any difference? (Clear cookies before attempts.)

I don't think that's applicable as I don't see it on my mobile. I think it's always on for smaller screens.

If the problems are happening on a WikiMedia wiki, try logging in on another wiki, preferably one that does not share a second-level domain name (so if the problem happens on xy.wikipedia.org, try for example xy.wiktionary.org).

It works fine on all other wikis I tried (meta, commons, Wiktionary). Only on MediaWiki site on my mobile, I'm not being logged in automatically.

I think I've answered the relevant ones. Hope I haven't missed anything obvious here. I'm not sure how to get the network requests on mobile.

Well, if you still experience it with Enhanced Tracking Protection disabled, since with ETP this is extended behavior. Although ETP didn't exist at the time task was filed.

I tested it, and works as expected on current Android Firefox stable (120.0): with ETP disabled, edge login works and I'm logged in immediately. With ETP enabled (default), edge login and autologin fail but top-level autologin works, so I get logged in without further interaction once I click on Login in the menu.