Well, you asked me a question about how the model behaves, and so my answer was of course written in the context of the model being correct, which it may not in fact be. As for my original statement, "the Higgs mechanism may explain why quarks and leptons have mass," what it means is that--take a flyer for a moment--assuming that SU(2) X U(1) is the correct symmetry to describe a unified electroweak force and assuming that it is spontaneously broken by the Higgs mechanism, lepton and quark masses follow as an unavoidable corollary. (Yeah, I know that didn't make sense to you on its surface, but pretend it's an opera: ignore the Italian and groove to the music.)
But you are asking a deeper question, now. The way I look at it is that nature is, at its core, consistent with mathematical truth. When you talk about "our mathematics", you mean, "our mathematical formalism", that is, our choice of symbols and our methods for manipulating them. Using "our mathematics", however, we do uncover real mathematical truths. These truths are universal, and they do constrain the possible behavior of reality.
Mathematical truth is more fundamental than reality; mathematical formalism is not. We attempt to use our mathematical formalism in such a way as to uncover those truths that are relevant to the description of reality. When we have a description of reality (say, a description of how particles acquire mass) then we can make ironclad predictions about how nature would behave in such a universe, and test to see whether our universe does, in fact, match those predictions. The Higgs boson is one such prediction.
Is mathematical truth discovered, or is it constructed?
Apparently, you would agree with those who say that mathematics exists to be discovered: in fact, mathematical truth is more fundamental than reality!
If pressed, I would probably say that physical reality is more fundamental, and that mathematics is something that we humans construct. Sometimes, if we are careful, our math adequately approximates reality.