I think this changed in the most recent Android app redesign. IIRC, it used to have the standard text selection pill that’s in other Android apps, but now it’s a slightly different visual style and lacks the web search option as you point out.
I use that feature regularly when reading articles to look up names, unfamiliar terms and concepts, etc. Now I have to do select, copy, switch to search app, switch back dance.
Went back to a build from March 2021 and the web view doesn’t have “Search” as a contextual option. Funny enough there is only an option to explicitly remove it.
Chrome vs Firefox have different buttons for “Search” so I’m assuming they have a custom implementation. This is what I’ll continue with as well and see what are the possibilities.
“Web search” was added as an option and it will open the system default browser via a search query intent. This could be moved to an in-app browser if desired.