I'm a tutor and I wouldn't be happy to do any of the "creative ways" skittlesandbeer suggests, not for free.
I teach around half a dozen adult men at the moment, about the same number of adult women and the rest are children or teenagers. Some of the men are older than me, some are younger. I can categorically say that I am never going to sleep with any of them, no matter how well I appear to get on with them in the lessons. This isn't just because I'm married and want to stay that way - it would be the same if I was single.
I'm sure most of my pupils, young and old, think I'm their friend. Many of them chat to me about their lives, and I chat a certain amount about the mutinae of mine. A lot of my pupils, young and old, are lovely and I genuinely like them.
But, and here's the thing to hang onto, OP, it's work. They go home, I get on with my family and friends. I don't want to hear from them or think about them on a Sunday afternoon or when I'm on holiday. They are not my friends. Teaching is a kind of acting job in a way.
What you're seeing, OP, is a good tutor making a student feel enthusiastic about the subject, and as though they are capable of achieving. I don't see anything to suggest more than that. If you're worried, my best suggestion is to book your own lessons with her, separately. Then you'll see why he's enjoying it.