We have looked into this today and performed various testing scenarios (with both upgrading and with/without the mentioned plugins). These issues don't relate to performing the upgrade itself, but rather existing JavaScript issues within one or more third party plugins are surfacing when those plugins are installed with Moodle 3.8.
In the case of mod_groupformation, the key error is that some JavaScript is not being defined correctly, which is causing an error that prevents other JavaScript executing on the page, such as that which switches between the site admin tabs. I will attempt to contact the plugin maintainer for that plugin today and let them know about the issue (it is also worth mentioning that plugin has not been updated since Moodle 3.5).
For Madhu and others experiencing this issue who do not have mod_groupformation installed, I would suggest some other installed plugin is likely not following the current JS guidelines, and triggering a similar bug. I did also test mod_grouptool, which seemed fine and did not cause the same issue. If you are able to identify the culprit(s), the best course of action would be to contact the relevant plugin maintainer, so they are hopefully able to address the issue and publish a compatible version.
Regarding the search.php URLs ending in #linkusers, #linkcourses etc, those are correct on the relevant admin section, and have just been updated in that way to allow refreshing/navigating back in your browser to that admin page to retain the correct visible section on the page, rather than defaulting back to the base "site administration" tab.
I hope that information is helpful, and you're able to find any remaining sources of the issue soon!
Mick