It looks like there are at least two distinct roots involved, one of which came into English via a couple different routes.
One root is the Latin
scala, meaning 'stairs, ladder'. This gives us the sense of climbing, and idea of a sequence of ascending steps also gives us the sense of musical scales. The idea of fixed intervals also gives us the sense of numerical scales and scales on maps. From that we also get the sense of size more generally.
The second root is the Proto-Germanic *
skalo, which could apparently mean 'scale,' 'bowl,' or 'cup'. It ultimately comes from a root that means 'split'.
The Online Etymology Dictionary says this comes from the notion of splitting a bivalve in half and using its shell as a cup. This old Germanic word was borrowed into French, probably via Frankish, to refer to fish or snake scales, and then it was borrowed into English.
This Proto-Germanic root became
skal in Old Norse, and in the plural it was used to refer to the device for weighing things, because they use pans or bowls on each side.
Coincidentally, that Proto-Germanic root also gives us the English word
shell.
