Haven’t checked how Bitwarden is handling favicons, though the easiest way is to “just” use /favicon.ico which is kinda standard though not every page ahderes to that especially as they’re other ways to define them within the HTML now.
So for example https://assure.ameli.fr/favicon.ico doesn’t exist.
The thing is this to get favicons from the HTML you would actually have to fetch it and parse it, easiest might be if the browser extension would take care of that and then sync it back but that would only work if a user uses any browser extension and only after the extension has been used.
And then also any of the HTML parsing ways would make it a little more complicated to when or how to keep it fresh, because they change at times…
On this note I can’t get my Bitwarden browser extension to refresh my favicon 
in desktop app it’s already refreshed, also tried removing and re-adding the domain and close my browser