For 10 years I had one uke. A no-name bought from an antique store in the 1980s. I only played it occasionally; mountain dulcimer was my main axe... and that's stretching the truth because really I wasn't much of a musician at all. Months would pass when I never touched an instrument.
In the 90s the bridge popped off. I went to the guitar store and bought the only uke they had -- cheap garbage. I might as well have wrapped a shoebox with rubber bands. It was that or travel to the one store in town that stocked Kamakas and pay a fortune. At that time I was only playing uke to amuse my toddler (whose rough handling of the old uke is implicated in the loss of the bridge) so investing in a quality instrument would have been absurd.
Life happens, and around 1999 I decided I did indeed want to learn to play music at a measurable level of competence. And I wanted a new dulcimer. Searching for dulcimers online, I saw a report from NAMM about the Magic Fluke company and ukes that cost a manageable price and (reportedly) did not sound like wet cardboard. I ordered one (it is still my main player) and discovered, as only a clueless indifferent musician can discover, that a good-quality instrument makes a heck of a difference.
Fluke became my one uke (the garbage laminate went to my son but he doesn't play it now, either). Then in 2007 there was an ad in the paper about a uke club, and I attended the first meeting, and suddenly I was playing and gigging weekly. And we were all playing each others' ukes. By this time the ukulele market had grown far beyond K-brand, garbage, Flukes or nothing. Banjo ukes are nifty. Resonators too. Warm-toned tenors, punchy sopranos, ringing cedar-topped concerts. I was discovering a whole new world of tone color -- after literally decades of not paying attention and not being a good-enough player to notice.
So now, yes, I own several ukuleles. 8, maybe 10. Offhand I lose count and I'm not sure whether to count the garbage laminate or my first uke (which I believe could be fixed by anyone with the right adhesive and clamps, which is not me). Some ukes it is not clear whether they belong to me or my son (he claims the Werco banjolele and might be the true owner of the baritone but don't tell him I said that).
Could I still get by with only my trusty Fluke? Sure. I can eat rice & beans all week. In fact, I love rice & beans (red beans with andouille, moros y cristianos, frijoles charros...) but I love other foods, too.
Sorry to ramble on like this. You asked! I think any discussion about this needs to be understood in the context of each player's own personal journey and where they happen to be at this moment along the path. Someday I might become such a serious musician, with such a strong personal style and focus on one type of music, that I pick one perfect uke and that's the only one I'll want to play. Hasn't happened yet, might never happen, but the future is bright and unpredictable. Wherever you're at, more power to you.