Is it necessary to handle every 404 reported in Google Search Console?
You only need to resolve 404 errors in Google Search Console (GSC) if it's not meant to be a 404. ie. If your site is erroneously returning a 404 for what should be a valid (200 OK) URL.
If the reported URL is not meant to be a 404 then you should fix your site so that the URL is no longer returning a 404 status (and the correct content) and then mark it as "fixed" in GSC. By marking it as "fixed" you are telling Google that it is no longer a 404.
If it is meant to be a 404 (as in this case) then just ignore it. And certainly don't marked it as "fixed" - because you've not "fixed" it and it's not meant to be fixed. The 404 is the correct response. If you mark a URL as "fixed" that still returns a 404 then it will simply reappear in the 404 report when Google next crawls that URL.
These errors are fixed for users but still there in webmasters. They re-appear as we mark them "fixed"
By "fixed for users", you presumably mean you've corrected how the links are generated on your site. The old (incorrect) URL that generates the 404 is still a 404 presumably? In which case nothing has been fixed as far as that URL is concerned and consequently, nothing should be marked as "fixed" in GSC. If you do, then they will indeed reappear in the 404 report, since they still return a 404 (which is the correct response).
Should we handle such URLs by redirecting to a common page (like home page)?
No. That's misleading for users and search engines. A "redirect" tells users and search engines that the page has moved to a different URL. There was never a page at these "wrong URLs". It's just a 404. Redirecting many URLs to a single URL (like the home page) is likely to be seen as a soft-404 anyway by Google and may still get returned in the GSC 404 report.
The unfortunate reality is that the 404 report in GSC can become huge on some sites - but that is really a fault with the 404 report, not with the 404's themselves. (Really, the GSC 404 report needs an "IGNORE" option, in addition to the "MARK AS FIXED" option.)