In the Standard Model properties are defined as relationships within/between symmetry groups. There are only so many things you can do to/with/in a symmetry group, and that's where the quantisation comes from.
But... that's a mathematical metaphor applied to observations. It's a good fit, but it doesn't explain why it's those symmetry groups and not others, or why symmetry groups are a good fit at all.
There's likely some kind of fundamental mechanism that generates these symmetries, and no one knows what that is.
Wikipedia suggests the quarks that make up the proton have charge ⅔e and -⅓e
Gemini advanced says it’s the latter, because of color confinement. But I’d defer to a human expert
We explain this by saying the quarks have a color charge and it must always be neutral. A single quark would be lets say red, but that's for some reason not possible. If we try to rip the quark out, it takes so much energy that this energy can be used to create another quark that results in a color neutral particle (red, antired)=meson, (red,green,blue)=baryon.
NB: this is a bit simplistic and other comments explain this quite in detail NB2: this color charge is just a name, its not an actual color