Historically, the name for a weighing device is scales. It is plural because scales had two clear parts in which one thing was weighed against another. So, they looked like this:
Modern scales don’t involve balancing things in two plates (photo from here):
(b)AmE has changed along with the scales, so that item (b) is usually called a bathroom scale, but scales is still used for the older kind. In BrE and AusE, however, it is still called scales, no matter whether it has two salient parts or not. When Anna Wierzbicka (Semantics: primes and universals) asked Australians why the word is plural, they answered that it was because there are lots of little numbers on the contraption. This seems to be a case of the word leading the thinking about an object. That is, because they say scales instead of scale, some people think about scales as being 'made up' of little numbers because they need to make sense of the fact that this singular object gets a plural name. Wierzbicka also notes that Australian English has shifted from speaking of a pair of scales, to a set of scales (for (b)). There, it looks like the name scales was broadened to cover (b) as well as (a), but when people started to think of numbers (of which there are many) rather than the plates on which measurable bits are put (which come in pairs in (a)-type scales) as the 'plural' part of scales, they shifted to thinking of scales as sets, rather than pairs.
Hm, aren't you just dying to take a lexical semantics course now? Or at least in the market for a textbook? Hey, maybe I can get you a discount...