For me, my first bass was an American Standard Jazz Bass. The Jazz Bass was also the one electric bass I had played before (a few lessons on the school Jazz Bass with my orchestra teacher - a great bass player, upright and electric). So, from the start, I've had a love for the Jazz Bass. It's probably a nostalgic thing - if I had started on a Precision, I'd probably be in love with that instead.
For me, I've always loved the aesthetics, the thin and narrow neck, and the tonal variety I can get from the simple-yet-versatile V-V-T setup. I love the simplicity of my Jazz Bass - and yet, in spite of this simplicity, I can do so many things with it. Lately, I've grown to have a new-found respect for the Precision bass and GAS very hard for one to match my Jazz. I dream of having a "matching pair" of Jazz and Precision basses.
Now, if I had started on a StingRay, I'd probably have been a StingRay devotee from the start. Same goes if I had started on a five-string. Now, however, I do own a StingRay5 and though I'm not sure the fifth string is still for me, I love the bass. It does other things my Jazz cannot - yet my Jazz also can do things the Ray cannot. They compliment each other in my book and I will always have a love for Jazz Basses, but I agree - there are a world of other basses out there, many with their own unique tone, so there's no need to restrict yourself to "just Fender basses."
I could, however, probably live with "just" an assortment of Fender and MusicMan basses (4s and 5s; fretted and fretless; Jazz, P, single-H, dual-H - all the combinations). I'd be a happy boy then.