A rain door is really just a shutter. In Japan, having shutters for rain protection is very common. When there were no glass windows, the shutters were simply like exterior doors in Japan, so "rain door" kind of makes sense as "shutter"!
This is a jukugo word, but it actually uses kun'yomi for both 雨 and 戸. Additionally, the reading for 雨 is actually slightly different from what you've learned, so here's a mnemonic to help you remember it:
Have you ever thought a shutter is like an armor (あま) for a window? They're both protections, and are usually made from metal. Kind of makes sense, right?
to renovate a shutter
to hit a shutter
to open a shutter
I open the shutters every morning.
Recently, there are fewer houses with shutters in Japan.
I woke up in the middle of the night because of the sound of rain hitting the shutters.