This new Apple Music feature just came out with iOS 16.2. I played with it for a second.
It works suprisingly well. I tried it on old songs in both Spanish and Hebrew. Impressive!
It looks like you can use Apple Music Sing on millions of songs. Which means voice removal is not a manual re-mixing effort but rather a clever automated process. Back in the day, karaoke software removed voices by filtering frequency ranges where most of the voice content usually falls. The result sucked, naturally.
This is another area in which Machine Learning allows computers to do things that were impossible for computers to do. I played with a Open Source TensorFlow based Python Library called Spleeter a while ago. It worked great.
I bet Apple is using something similar. I would also bet Apple is using ML for Speech Recognition in order to get the timing of the lyrics just right on songs that support this feature.