I think the even better analogy than browsers is search engines. There aren't any network effects or platform lock-in, but there is potential for a data flywheel, building a brand, and just getting users in the habit of using you. The results won't necessarily turn out the same - I think OpenAI's edge on results quality is a lot less than early Google over its competitors - but the shape of the competition is similar.
There is no moat
On iOS with the Apple agreement, and on Android (though the question of hardware remains when considering beyond Pixel phones).