The question invites generalisations that I'll try to avoid and just speak from observations from my environment.

I have noticed that those who dance well, especially to tunes that may be unfamiliar melodically but with familiar rhythms, tend to pick up the lilt or swing in a tune rather quickly. Your daughter may (or may not) have shown this ability; she may yet be too young to have exhibited it. If she has, she may be already on the path to 'internalising' the music she hears and plays.

Most formal music teaching requires students to learn intervals aurally and be able to reproduce them orally; your description seems to indicate that this is what's happening and I wouldn't be too concerned about where it might or might not be heading.

Nine years old is an impressionable age (like most) and, at that age, I was just enjoying music and being able to sing. While my daughters (at that age) were learning instruments formally I regarded it as more important that they also "enjoyed" the activity. To this end I wasn't too fussed about missed practices and ensured they came to festivals and gigs and experienced other people enjoying playing and being actively involved in music; they're both dancers as it happens.

This has had the effect of motivating both of them as players (piano for one and recorders for the other). Both can sight read reasonably well and they both have good ears but, as yet, neither plays by ear unless they're engaging in displacement activity. This occurs when they know they should be preparing for a particular event and they pull out my concertina or the fiddle or ukelele (they haven't yet bothered with the banjo but have expressed interest in the future) and spend time working tunes out by ear. I am confident that, one day, they'll do well on any instrument they choose.

I did ask the recorder teacher if she could recommend any of the violin teachers in town but, describing my recorder players strengths, she said none of the violinists had the approach that my daughter required; I was confident she was levelling with me and I left it at that. There's time enough in the future for both of them to grow into whatever they want.

But I'd probably suggest that, if they do want to get into fiddle, they seek advice on basic fingering and bowing techniques from both formal teachers as well as role-model practitioners.

Cheers, Rowan