No, being able to solve a cube requires no mathematical aptitude whatsoever. It just takes a modest amount of memorizing sequences of actions. Even developing your own entirely intuitive solving method doesn't require "mathematics", per se, just a lot of pattern recognition, cleverness, and a bit of intuition. To be fair, that sort of activity is more like doing mathematics than it is any other academic subject, but that's not because puzzles require math, it's because math is full of doing puzzles.
The reason you hear people say this is that the sorts of people who are attracted to mathematics and the sorts of people who are attracted to puzzles overlap pretty heavily. And anyone can do either of those things if they try. Being interested in both = trying in both = succeeding in both = bystanders saying "oh look another math guy with a cube". That's all.