Every time I fall for a student (and there have been a few I wanted to bundle up and take home), I start dreaming about how cool that will be when we get to her senior year in high school and I'm the one giving her the flowers at the spring show. (These kids are 4 or 5, mind you.) Every coach wants this.
While kids get attached to coaches, they tend to transition pretty easily (insultingly easily) from one to the next. It's the adults--both adult skaters and parents--who get convinced that they (or their child) can work with one coach and one coach only.
I've seen this manifest in parents who won't let kids take classes or seminars with another coach, even with a guest coach, because "oh, I only learn well from Coach Svengali."
This can become difficult if the coach's life changes and they can't coach you anymore. I've seen parents berate coaches for moving, switching rinks, and having a baby (what do you mean you won't be teaching for 6 weeks!).
As I said, coaches really like the idea of taking a kid from tot to the Olympics (think Brian Boitano and Linda Leaver), and especially at the recreational level you see this a lot. My daughter has managed to have the same dance coach (slowly morphing into a dance partner) from Preliminary through Internationals (she's on her fourth or fifth, I've lost count). But you need to let this happen naturally, and not out of misplaced ideals.
You stay with a single coach because you have a positive relationship, work well together and are reaching the goals you set. If you're not passing the tests, or doing well at the competitions, or learning the jump with that coach, then either change your goal (to I love this coach and don't care about that other stuff), or switch coaches.
And of course, you have to bring it up: what happens when parents believe only one person is qualified to coach their kid and the coach becomes self-obsessed, abusive, and capricious? I've seen coaches who have so convinced the kids and parents that they can never switch, that in lessons they ignore the kids or use abusive language and methods. The parents, and skaters, start learning that the choice is not Coach Svengali or Coach What-a-Relief, but Skate-Or-Quit. Quitting is always an option, but not because of my-way-or-the-highway coaching.
Sometimes you get lucky, and stumble onto a coach that really is the only coach for you. But make the choice for positive reasons.
Have you or your skater worked with the same coach over a long period?