Floral patterns have been popular in Hawaii for a long time, but modern "Aloha" wear mostly stems from products created by Ellery Chun, in the 1930s, who used the fabric remaining from the production of kimonos to create shirts.

The shirts caught on very quickly, selling as souvenirs to tourists and later to American servicemen and their families stationed in the islands, and soon there were a lot of copycats, both in Hawaii and elsewhere.

The adoption of the shirts (and other "aloha" patterened clothing) as more formal wear stems from a sort of image campaign by the city of Honolulu, allowing city employees to wear the clothing in order to promote Hawaii as a relaxed destination and yet still one in which serious business could be conducted.

This became more and more of a "thing" until it was essentially public policy throughout the islands.

The question is best answered by Dressing Down: Modernism, Masculinity, and the Men's Leisurewear Industry in California, 1930--1960, a dissertation by William Ramsey Scott. He starts to discuss it on p. 110.

Hawaiian shirts were invented in the 1930s by a Hawaiian, Ellery Chun, who sold them to tourists. They were based on traditional Hawaiian kapa or tapa cloth designs, reinterpreted on silk or rayon made up in the style of the checked palaka shirts worn as work dress in the islands. After WWII, Hawaii's popularity on the mainland gave the shirts more cachet, and they were frequently worn as resort wear.

It should be noted that the original textile patterns were highly influenced by the kapa textiles, but Robert C. Allen points out in Creating Hawaii Tourism that the designs we use today - usually featuring realistic landscapes, plants, etc. - owe more to designer Elsie Das.

I'm afraid I don't know exactly how the Hawaiian shirt is worn in the Pacific Islands, so I can't really speak to that.