No, at the moment all pages/sites get the Voice megaphone guy (which I did recently update to have a retina-resolution icon).
It could be good to be able to change it though, for sites with a strong brand that they want to maintain. Especially as some search engines (e.g., duckduckgo) also show the logo in the search results.
It's not just as simple as using the site logo though, as there are strict size/resolution requirements for the icons. There'd need to be some custom processing of the logo for it to work. I'll look into it.