Yes. It's something you could say after someone has recovered from any illness or a surgery.
I guess (only guess) that it's because fit not only mean healthy but also more or less thin, and violin (fiddle) is thin, it just came from one meaning of fit to the other. That's my guess. I'm bad at explaining, sorry ^^; But I've always found that saying at least weird.
American English sometimes differs from that used in the UK and even within the UK you get variations. Some of these strange expressions (healthy as a horse, fit as a flea) are used only in certain regions. And we Brits can find US phrases confusing - especially if we don't watch a lot of imported TV!
Well, you made me laugh x) but health is no joke :o so I can only wish your goats even more health ;)
That would be me x) that will be my saying of choice now! Thanks for sharing!
Same in Polish. In Polish you can also say healthy as a bull, it kinda makes more sense than a fish, from non-scientific point of view at least ;)
In Italy we have "sano come un pesce", which is a perfect literal translation of "healthy as a fish".
Correct UK English would indeed be "as fit (healthy) as..". American usage might be different.