Just the usual marketing angle, IMO. It's not TV, it's HBO.
No one is reluctant to use the word smartphone to include iPhones. I don't think anyone is going to use the Apple Intelligence moniker except in the same cases where they'd say iCloud instead of cloud services.
It's also a little clunky. Maybe they could have gone with... xI? Too close to the Chinese Xi. iAI? Sounds like the Spanish "ay ay ay." Not an easy one I think. The number of person-hours spent on this must have been something.