iPhone 8 that is the oldest supported for iOS 16 has an A11 chip; the second gen 4K has A12. The first gen 4K has an A10X chip - maybe it needs the "Neural Engine" that first appeared in A11 but the 2nd gen does have it.
In addition, the Apple TV is plugged in, it doesn't have to limit its power for battery reasons.
Edit: per the bottom of the page...
> Apple Music Sing will be available on all compatible iPhone and iPad models as well as the new Apple TV 4K.
:/ maybe they do ML to process stuff on-device, per other comments' speculation, that justifies requiring the newer model?