Presumably the same way we distinguish individual quarks: by smashing the atoms up.

(The more interesting question would be the opposite: what if it was two electrons per proton? Then you could throw around some photons and end up with a half-proton negatively-ionized molecule. What would that look like?)

You are going down the path of theoretical particle physics! It is the ultimate question of that to answer what is the fundamental element that makes up matter and what should we "name" that has a useful property that can either be used or helps to explain how other things work.

In reality, "protons" do not "exist" but are semi(very) stable collections of energy that interact in an interesting enough way in a group that it is useful for us to retain the name, rather than refer to it by its constituents.

Electrons don't really glob up into things like atoms due to repulsion (no moderation by the stron/weak nuclear forces) so we don't have a really useful reason to keep going beyond the definition of the electron so we just stop trying to find additional constituent parts.

We see naked protons and mismatches between nuclei and electrons all the time, so I don't see why half-charge protons would be "_always_" paired off.