Is electric charge a physical property of matter?
Yes, charge is a property attributed to fundamental particles.
To quote Wikipedia:
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field.
ALL particles are either composite or elementary / fundamental. These terms are rather self-explanatory: whilst a composite particle is composed of other particles, an elementary particle is fundamental in the sense that it is not composed of other particles. As you can see from the above diagram, three groups are labelled as fundamental: these are quarks, leptons, and bosons:
After this, all other particles are composite particles (with the exception of antiparticles, hypothetical elementary particles, and the Higgs boson), because we know that they are composed of other particles. Observe that protons and neutrons are composite particles , specifically baryons, that are made up of quarks:
Returning to the topic of elementary particles, then, charge is a property that we allocate to said particles based on our observation of how they behave in an electromagnetic field. This of course reflects how different particles interact with one another via the electromagnetic force. But outside of this, what is charge?
We can say that protons are positive particles, but then we ask the question why? And of course we know that we can look into the substructure of the proton in an attempt to explain this, and if we do then we will find quarks that have their own charges. Starting with any magnitude of composite particle, in an attempt to explain charge you will eventually come to the elementary particles, and by that logic you will be unable to continue.
This is because, as stated in the answer, charge is simply a property of these elementary particles, alongside mass and others.