If you want to embed an HTML client somewhere else then there is not a NIP for it because it's not advisable at all, but you would have to use an <iframe> or some other way to render HTML in that other environment. And then you could inject a faux window.nostr global in there.

Not the same thing, but as an inspiration, https://github.com/fiatjaf/window.nostr.js injects a window.nostr global in websites, but instead of an extension it powers that with NIP-46 calls to bunkers.