Do people in Japan really shower all the time after a rainstorm, or is that just an anime thing?
Regardless of whether the rain was cold or not, taking a shower after a downpour has a health related reason. The rain is not a very clean source of water with the factories and other pollutants in the air. Taking shower was meant to wash away the dirts.

As to offering shower to non-household members, it is a show of hospitality. As Dimitri MX and you said in the comment, it would differ from one people to another. Some would offer shower, while others might offer just some towel.

As a comparison, in Indonesia (another Asian country), we were taught to have shower, or at least wash our head after we got wet by rainwater. The believe was that if you let it be, you will get headache after some time.