@ma_ko, welcome to the forum! 
@ppl , welcome back!
As far as I know, Bitwarden uses the information on a page to fetch an icon that is stored on Bitwarden’s server. The apps access the icons on the server, not from the websites directly.
Sometimes, Bitwarden doesn’t pick out an icon like the web browser does, and it’s most likely because it uses a different rule (different web browsers also seem to use different rules) to select an icon.
What may work sometimes with Bitwarden is to input another URL with a different hostname, for example, web.domain.com (you can also look for ideas in the website’s page source) instead of www.domain.com, to see if it has different icon placements that would match Bitwarden’s picking rule. Beyond that, it’s basically either living with it or reporting it to customer support. See the example here: