If they try this, and I'm not convinced they will, it won't be a general purpose search engine. It'll be something that only runs on their devices and in their ecosystem.
Apple does have a crawler of its own, which feeds into Siri's knowledge. But, there's no public front end to use it as a search engine, though there was up to about 2021 or 2022, provided as a bit of fun by some enterprising third party.
Spotlight is already their search engine. You can search their app store, movies catalog, the web, etc. They're already doing this.
I don't believe that would be possible. At worst, it could be locked behind an iCloud authentication.