Is there anything stopping Google from just making up their own TLDs for their browser?
Back in the 90s AOL did similar with their browser and it was fairly common to see companies list their AOL Keyword in advertising.
I think it's unlikely to happen too soon though as they generally own the search box in chrome and can autosuggest whatever they want when you enter a term anyway which is probably the contemporary equivalent anyway.
There are various fake crypto/blockchain-related fake TLDs you can add to your browser through browser extensions, but the risk of future name collisions is very real. And a domain name that doesn't even work in other browsers isn't useful at all. Having something as basic as "Share URL" not work results in a fundamentally broken (and bad) user experience.
This would not be a concern because Chrome has a massive install base. Similar to .onion, no one would register or potentially be able to register the one Google would adopt.