You can produce a Hawaiian tune on a ukulele. You just have to use an alternative tuning on either a soprano or concert ukulele. Original Hawaiian ukuleles were most often tuned ADF#B. With today’s larger ukuleles if you tune a soprano or concert ukulele to A4-D4-F#4-B4, this slightly higher key will make a brighter, clearer tone and give the ukulele a more traditional Hawaiian sound. You would likely not want to try a Hawaiian tune on a tenor or baritone ukulele.
It should be remembered that the original Hawaiian ukulele is smaller than the ukulele that you can find in the market right now. This means that the sound is a bit different. The type of wood used is also different from the type of wood used in the ukulele that you can buy in the market right now.
For example, the ukulele from Hawaii is made of Acacian Koa. The wood used for the ukulele you can purchase outside Hawaii is mahogany. The change in wood alone can already make a difference with the sound. To make your ukulele sound more Hawaiian, you can change the tone of your ukulele a bit.