As you go up the periodic table (more protons), the ratio of neutrons to protons steadily increases as well. Are we sure there are absolutely no protons and electrons in a neutron star, or could there be so much more neutrons that we cannot measure any protons and electrons? Perhaps then a neutron star is a nucleus of some huge element with a neutron:proton ratio higher than we can distinguish.
To be considered an element, they would have to have a positively charged nucleus. They don't. The neutron star is largely neutral.
They would have to have a cloud of surrounding electrons. These electrons would have to share "orbitals" with other electrons around other, nearby neutron stars. That doesn't happen.
Finally, at these scales, gravitational interaction predominates. Even if neutron stars did have some positive charge, and did have electrons orbiting them, interaction with other neutron stars would still depend a lot on gravity.
Finally, even if a neutron star was charged, it's not clear whether electrons around them would actually fall into some kind of orbitals governed by quantum mechanics. QM tends to not happen at such a scale.
In conclusion, no, they are completely unlike atoms and their nuclei.
Neutron stars are not to be considered as an homogeneous object, they have different properties at different layers which are dependant on pressure and temperature for example. Thus in the core below some critical temperature you could have superconductant protons (or any charged baryon) thus meaning you would not find atoms, but more like a soup of free particles. In conclusion I don't see how could this fall into the definition of a chemical element.
Reference: Neutron Stars 1: Equation of State and Structure. By P. Haensel, A.Y. Potekhin, D.G.
The 'stuff' comprising the portion of a neutron star theoretically comprised mostly of neutrons has been referred to as "neutronium" in various sources, one of which is science fiction.
I have always found imagining what this 'neutronium' would actually look like to be an amusing exercise. However, attempting to define this 'stuff' as an element in and of itself is no different than performing the same attempt at categorization for an exclusive mass of protons or elections. None of these components define atoms or elements singly.
Think of a neutron as a proton + electron pushed into it. Neutron stars are a lot like that; any electrons that have met a proton will be very strongly forced into forming a neutron. This happens because an object containing very many protons and electrons has collapsed below the electron degeneracy pressure limit. Presumably, there weren't exactly the same number of protons and electrons before the collapse; and a fraction of those particles did not find their counterparts before the collapse stopped. It's thus quite possible that a particular neutron star has a small positive charge as a whole, and it probably contains some protons and electrons (not counting neutrons).
However, being composed of protons, neutrons, and electrons does not make matter an element. Being composed of protons and neutrons does not make matter an atomic nucleus. The forces that hold the neutron star together are unrelated to the forces that hold together an atomic nucleus, and its properties are very different, too.
A positively charged storm cloud might have many more protons than electrons and still not be called an "element".