Google's John Mueller repeated advice from a while ago that you should not switch your m-dot URLs to be the canonical URL, even though Google is fully switched over to mobile-first indexing. He said this is just because it is how it is done, how it was done, and switching it on Google's end would cause many large sites issues.
As a reminder, Google said this in 2017, "No changes are necessary for interlinking with separate mobile URLs (m.-dot sites). For sites using separate mobile URLs, keep the existing link rel=canonical and link rel=alternate elements between these versions."
John repeated this saying on LinkedIn, "Since Google indexes the mobile URL instead of the desktop one, should sites with m-dot URLs switch to canonicalize to the mobile version now? Tl;dr: no, don't change it."
He then explained why - in short, because it was done the other way forever, changing it, would cause some really large sites a lot of issues:
It could make sense: if Google is picking the mobile URL as canonical, shouldn't the site do that too? (Again: don't.) First off, if you have the time and use separate mobile URLs, then I'd suggest working towards a responsive design: using the same URLs makes things so much easier, even if it's just for some parts.If we started from scratch, canonicalizing and indexing the mobile version would be reasonable. However, *switching* canonicals is very hard, you wouldn't be able to trust any canonical links for a long time (some are Mobile->Desktop, some Desktop->Mobile), there would need to be a new "link rel alternate desktop", and all search engines would have to adjust. So, just keep it as-is (canonical means they're equivalent anyway), or take steps toward a responsive design.
FWIW by "canonicals" I mean the "link rel=canonical href=URL" elements in HTML or in HTTP response headers. Image unrelated, but technical SEO = gears, right?
When he was asked why sites aren't doing this right? He said, "I hope there are very few new sites doing this, but changing the infrastructure in bigger sites (like Facebook or YouTube, who I think both use m-dot) has got to be much harder than me doing posts here."
John also added this about the vary header in the comments - which actually was something I didn't know fully:
And a random anecdote - while checking this with the mobile indexing team, we realized that Google doesn't use the "vary" HTTP headers at all for understanding the mobile/desktop relationship. These are unnecessary for SEO (and we'll make that a bit clearer in the documentation). They're purely for usability, to help with any HTTP caches. You don't need to remove them, they're just not an "SEO thing".
Forum discussion at LinkedIn.
