Google's John Mueller confirmed that Google Search is working on an issue with some favicons not showing up properly in Google Search. The issue has been ongoing for weeks (around September 18th), well before the new site name and favicon release was announced, by the way.
John called this a "niche" issue, implying it is not too widespread. But I am not sure the definition of "niche" because I've been hearing about this issue for weeks.
Stefan shared an example of the Google Store having this issue, and yes, I can replicate it:
He added on Twitter "we've been having issues with favicons not being displayed for months now and ironically Google's Store is in the same boat. Have you seen any of this?" Yes, yes, I have.
There are also some examples at the Google Webmaster Help Forums of this issue.
So I asked John Mueller who replied on Twitter "We've looked into it, and the team has been working on resolving this, but afaik it's more of a niche issue."
We've looked into it, and the team has been working on resolving this, but afaik it's more of a niche issue. (Also, I wouldn't assume that Google sites get SEO right automatically, we're people too :-))
— ⛰ johnmu is not a cat ⛰ (@JohnMu) October 19, 2022
So if you see your favicons are not showing up properly, and you've done everything as per the requirements then hopefully Google will fix the issue soon.
Forum discussion at Twitter.