Gift giving should always come from the heart. BUT since you asked.....
How long have you been with this instructor? Just started, I would say maybe not. Been there for a year or more, like what you are learning - why not a gift card to the local coffee shop or Subway? If you like to bake, some cookies would be nice.
Just a few months in private lessons from her. But before that we were mainly learning from DVD's or from the community college, so we picked up a lot of bad technique prior to the private lessons. So we're really appreciate what we've learned and how we've progressed.
I work in a studio so I see all the gifts that come in for the instructors. It is really up to you, gifts are not expected but very much appreciated. I would suggest something consumable. If you know she likes coffee bring a Starbucks with you next lesson with something to eat. Instructors are always hungry. We makes pots and pots of coffee in our studio. Wine is good. Flowers are also appreciated. They stay in the studio where the instructor can see them all day. Something that can be shared, like chocolates is good because some instructors get more than others and this way they can share the wealth. I have never seen cash given ever and since instructors are professionals, they would be insulted. A gift card is better. Stay away from personal stuff like jewelry, clothes etc.
Ok, maybe insulted was too strong a word, awkward definitely. I have never seen money given, in my limited experience of 10 years working in a studio (Canada). So maybe it is common elsewhere but I honestly haven't seen it EVER. Perhaps it is the way we do business, I take care of payments, rarely do the students pay the instructor directly so to suddenly hand them $20 would be odd for everyone. As for your 'interesting thought', you are comparing apples to oranges. If a company I work for gives me a bonus, it is for performance. The company has clients. Often the company gives gifts to clients as a thank you for being a client. We should be giving our clients (students) gifts, not the other way around. The dance studio is the employer of the instructors, the students are the clients. If the studio decided to give the instructors a bonus, they would happily accept it. The clients pay us for a service, there is no reason why at Christmas, they should pay extra under the guise of a christmas gift.
I am an independent contractor. Most of my regular private students give me Christmas gifts- even the Jewish ones. I also get birthday presents and random surprise gifts. The gifts are always very thoughtful and much appreciated. The $ value is all over the place- from a hanging flower vase made from a recycled light bulb and string to custom diamond jewelry. I think anything would be ok to give, especially if it's coming from a couple.