Yes, singing, acrobat and magician are all hidden skills, which remain after you quit their respective jobs. It's easy to see if you use Twallan's Master Controller mod.
The only effect the hidden skill has is in which venues you can perform in, so for example I raised my acrobats job skill quite a lot of levels simply by raising athletics, to the point that I was a level 10 acrobat, but had a hidden acrobat skill of only 6. This meant that even though I could go for auditions at the big venues, they always turned me down and told me to come back when I was more accomplished.
So auditions go off hidden skill, rather than job level.