It rained on my island once. Rain is extremely rare. Mostly, it's just clear or cloudy skies.
If it rains, the music stops entirely.
I'm pretty sure it's actually because you're on an international island. Many games have very small differences to them depending on the region, and I wouldn't be surprised if there are copies in which your island can rain.
Hey! That's me behind you!
That was the first time I saw rain on the island too