OAI has certainly positioned themselves culturally the same way as Google did for search engines. Google this, Tweet that, ask ChatGPT.
We know now how much actual competition[0] Google had after the dust had settled, in all practical terms - zero. Even after all the SEO spam and enshittification they haven't lost any notable market lead.
Time will tell if ChatGPT ends up that way but unless OAI implodes (which isn't all that unlikely) they're on the way there.