Since a neutron star is basically just a vast amount of densely packed neutrons, I was wondering whether those neutrons form a single "atom" (of atomic number zero) or whether they are further apart and in a QFT-sense isolated?
It is a misnomer (at best) to characterize a neutron star as all neutrons. There are protons and electrons too.
Imagine compressing a bunch of regular matter at some point it requires less energy for a proton and electron to combine to form another neutron rather than for the electron to try to fill a very high energy state. That means there are so many electrons so as to fill all the states with energy up to the difference in energy of a neutron compared to a proton plus an electron. So those electrons stay around, as do an equal number of protons.
So if you want to think of it as an isotope, it wouldn't be a Z=0 isotope. And if it is so large you could even have protons entering one side and leaving another in a spacelike separated way, there is no clean sense where it is a single object with a certain Z. It is a system of many interacting parts.
Even if it is pure neutrons, I doubt that it can be called one single atom. Strong forces are short ranged and I don't think a macroscopic number of neutrons can form one single bound state. Instead they will form into many bound states each with a few neutrons.
Calling a neutron star a giant $Z=0$ atom would be like calling the earth, which is made of electrons, neutrons and protons, a giant atom with a giant $Z\neq 0$.