Once again, I am way out of my league in answering this. I may be wrong about many things here, comments appreciated
That was just a definition of mass. The Higgs explains where rest mass (but not gravity) comes from in a mathematically rigorous manner.
One of the attempts to explain how our universe works in a mathematically rigorous manner is the Standard Model. It tries to put all the fundamental forces (except gravity) under one umbrella, along with the particles in a comprehensive theory that explains the subatomic world in a consistent manner.
The theory, in the course of its evolution, has predicted many particles--including many of the quarks, the $W$ and $Z$ bosons, and of course the Higgs boson. All except the Higgs{*} have been experimentally confirmed.
All of these particles are necessary for the theory to work. The Higgs is a product of a neat mathematical trick (spontaneous symmetry breaking), that leads to the concept of "mass" blossoming into existence.
If the Higgs is not found, the whole theory does not work (or needs significant tweaking). IIRC, the SM initially predicted a massless $W$ particle and had inconsistencies--which were resolved by introducing spontaneous symmetry breaking. The original intention for introducing spontaneous symmetry breaking (and thus the Higgs), was to "fix" this issue in the electroweak interaction. From this point of view, the "Higgs makes particles massive" is more of a side-effect, an afterthought.
Mass is not exactly comparable to charge. Mass has two aspects--the gravitational aspect and the inertia aspect. EM forces are the "charge" analogues of gravity, but there is no such analogue for inertia.
Now, the Higgs is explaining the inertia aspect of mass, of which there is no electric counterpart. So there is no need to assume the existence of an electric analogue of the Higgs.
As for EM forces, they are already explained by the SM (the force is mediated by photons). The SM specifically leaves out gravity, but there is a hypothetical particle, the graviton, being researched--this may explain gravity as well.
As @David has mentioned, the above section isn't exactly correct. THe reason we don't need another particle for charge is that charge is already explained by the SM, mathematically (whereas, without Higgs, rest mass is something we just assume)
*As of today, this has probably changed--a particle that is similar to the Standard Model Higgs has been discovered by CERN