While a number of Google Search features are impacted by Google's core update, favicons showing up or not is not one of them. Yes, your site can drop in the rankings, or not show up in Google Discover, or be removed from Top Stories and other features but favicons are not impacted by core updates.
John Mueller from Google said this on LinkedIn after someone complained that their favicon disappeared from the Google Search results interface, John said it was not the core update that did it.
John wrote, "That would be unrelated to anything around a core update."
He then suggested that the user review Glenn Gabe's article on debugging your favicons in Google Search.
I am not sure if I covered this before, but I think this is somewhat known - that core updates don't impact the display of favicons.
Forum discussion at LinkedIn.
