Is there something about statistics that lends itself to this sort of saying, or is it just that people will say anything to support their case, and this includes citing irrelevant or incomplete statistics?
Statistics is about inferring something about a population, and that requires some level of interpretation.
More intuitively, "is the glass half full or half empty?". They both mean the same thing, but may have a different effect on the person who hears it.
So I would say it's the interpretation aspect which is the problem
P.S. There's an interesting article on the BBC website which may be worth a read.
P.P.S. If you meant this more generally, then there could be a case for saying that the frequentest approach to statistics can give a different result to the Bayesian approach.