The history of the 'ukulele is fascinating. It flows across oceans and back through time as far as you want to take it. Some people want to pin it down to absolute statements like "the 'ukulele is a Hawaiian instrument" or "the 'ukulele is not a Hawaiian instrument." I think this misses the whole point. The invention of the 'ukulele was not an event; it is a process (which continues today).
Musical instruments do not often spring to life fully-formed like Venus on the half-shell. For the most part they evolve gradually. They may pass through many different cultures on the way. The Hawaiian 'ukulele has antecedents in Madeira. Those instruments had antecedents on the Iberian peninsula. Which in turn had antecedents in the Middle East.
You could say the 'ukulele is a small 4-stringed oud invented in Arabia. That would be, historically, correct. But it loses any sense of what the 'ukulele actually exists to do in its cultural context. People get caught up asking "what is this?" when the real question is "why is it this and not that?"
An instrument is more than simply its characteristics as a physical object. How is it tuned, who plays it, how do they play it, what kind of music do they play on it, and on what occasions? What role does this instrument fulfill in society? A church organ and a harmonica use similar mechanisms for producing sound, but they are not the same instrument.
A fiddle and a violin are not the same instrument, either.
Madeira did not have 'ukuleles. Madeira had other instruments, brought some of them to Hawaii, and started (or I should say continued...) a process that lead to the 'ukulele. Isn't that great?