Google’s John Mueller said Google does support ETag/If-None-Match after this Search Engine Land article shows that Bing supports it. He said it is rare to see sites using it, most use If-Modified-Since instead.
John Mueller when asked if GoogleBot supports that ETAG, said last month on Twitter “Last I looked we supported that, though most sites seem to use If-Modified-Since more.”
Last I looked we supported that, though most sites seem to use If-Modified-Since more.
— 🍌 John 🍌 (@JohnMu) November 13, 2018
Mozilla has a more technical description of this header.
Forum discussion at Twitter.