Firefox 101 Snap Linux:
(settings - Kagi is the only one without icon)
(search bar - Kagi is the only one without icon)
What could be the problem? Taking https://github.com/parski/wolfram-alpha-firefox-addon/blob/master/src/manifest.json as an example that worked, and comparing with the favicons from DuckDuckGo, Google, Bing, Yandex, you will see that Kagi's favicon is the only one that will try to download the file instead of just displaying it. Firefox probably made some bigger mess that won't work with these types of favicon URL in the manifest.json.
Possible fix: Change the current header "Content-Type" from "application/octet-stream" to "image/x-icon" like everyone else is doing.