Well, I never owned one of them, so maybe I'm not supposed to comment on this. But their website was very clear: The OS on the first iPhone was OS X.

https://web.archive.org/web/20070112064939/http://www.apple....

It's about as Mac OS X as Apple Watch runs macOS.

It's not the same as Mac OS X or macOS, they share a common kernel, system libraries etc but the userland is very different on an iPhone or Apple Watch.

Apple advertised the first iPhone to run OS X: https://youtu.be/VQKMoT-6XSg?t=506

It's not the same version of OS X that ran on Apple's computers. The "it's OS X" was more for marketing, they just share the same "core".

You could argue that the iPhone currently still runs macOS if you used the same definition today. They share kernels (iirc Apple always kept the ARM patches to Darwin closed-source), BSD-based userlands and the iPhone used versions of the macs application libraries.

A big difference is the iOS and macOS use different compositors.