For example, the Sun is a giant sphere of positively charged plasma. About 72% of the elements in it are protium. This means that these protium atoms would have had to lose their electrons, right? If so, the majority of the Sun would be protons. So just as there are neutron stars, would stars like our Sun be considered "proton stars"?
This may be a stupid question, but what I'm really getting at is: are stars mostly protons? Is what I described how it all works?
Protium is a proton + an electron.
Under enormously high pressure, it's energetically favorable for electrons to merge with protons and become neutrons - see here.
are stars mostly protons
By mass, yes, at least before they get too old.
The mass of the universe is more complicated, but anything solid that we think of as matter is made of atoms, which are by mass, mostly protons and neutrons (you can break it up further than that if you like, but that's best for another question).
Most hydrogen has no neutrons so any hydrogen rich object (the sun, most young stars, gas giant planets) are by mass, mostly protons. That's no longer true when a star gets close to the end of it's life and has burned much of it's hydrogen.
Jupiter, by mass is roughly about 80% protons. The sun, because it's been turning hydrogen into helium for about 4.5 billion years, is roughly about 67% protons by mass. The Earth, mostly other elements, Oxygen, Silicon, Iron, etc, is a about 50% proton by mass.
Would stars like our Sun be considered "proton stars"?
I suppose you could use that term, but I don't see any benefit to it. It's not too different than calling the sun a "hydrogen" star. All stars start out as hydrogen stars.
The analogy is false.
Neutron stars consist mainly of neutrons, with a small fraction of protons and electrons in equal numbers. The fraction is density dependent, but is in the range 1%-10%.$^{*}$ Neutron stars are so-named because neutrons dominate both by mass and number.
A "normal" star consists of about 75% Hydrogen and 23% Helium and 2% heavier elements (by mass; the fraction varies with age, mass and initial chemical composition).
Most of this material is completely ionised, so the hydrogen is in the form of protons and an equal number of electrons. Whilst protons (just) dominate the mass, you cannot call it a proton-star because there are equal numbers of electrons. Actually, because of the contribution from ionised helium and heavier elements there are more free electrons than there are free protons.
$^{*}$ Neutron stars also have a crust of neutron-rich nuclei (along with enough electrons to balance their charge), but that is a small fraction in terms of their mass and in terms of particle numbers, so is unimportant to this argument.
