A later development.

kiša < *kysja ≃ PSl. *kyslŭ 'sour'

It looks like a semantic widening of words meaning 'to become sour' to 'get wet, soaked'.

kvasiti1 – wait for the dough to rise and become sour
kvasiti2 – to wet, moisten, drench

kisnuti1 – to become sour
kisnuti2 – to get wet from the rain

I noticed for a while that you are linguistically knowledgeable .  I have no idea but you may know why we say "дожд".