Is it possible neutron stars are actual elements?
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.