I feel this is overstated somewhat. Any two ukes are going to sound a bit different, whether one is a vintage Martin and the other is an off-the-rack Luna, or if both rolled off the assembly line the same day at the same factory.
There are so many variables. I believe it's clear that generally speaking a handcrafted uke made of solid woods will be sweeter and more resonant than a mass-produced laminate from China.
Differences in woods, build, etc. will also be more evident in acoustic instruments than solidbody electric things.
There is so much you can do to get many sounds from one ukulele. I don't believe in one trick ponies.
Even a simple string change can have a marked effect on the sound/tone of any uke.
Also, technique. Your fingering, hand placement, attack, etc. All these things play into the overall sound(s) you get from your instrument.
Obviously there are cases where a uke just doesn't work for you, or please the player. Bad intonation, thick ply and glue that dampens projection, etc.
But in my experience, overall, I feel I have learned more from trying to work within limitations. In other words, I no longer have leisure cash to try dozens of ukes out. Rather than take the approach of "Maybe a cedar top will sound richer" or "The tone I want can only come from a solid mahogany uke at least 50 years old" I just try to get the most out of what I am fortunate to have.
I say these things because I think it can be disheartening for a new player who gets excited to hear Jake S or Ohta-San the first time to think they have to rush out and spend a grand or more to emulate their heroes.