Any package management tool can be used in any distribution. So any package management tool is distribution agnostic then. Fedora had 4 system package managers at same time: apt, yum, smart, dnf.
Sure, with enough work, you can probably make Debian work with pacman, and Archlinux with apt-get. The problem with those package managers is that they install everything on /. If you try to use both package managers at once, they'll most likely conflict with one another.
Nix on the other hand will install each package under it's own version- or build-specific directory under /nix. ~/.nix-profile will be a symlink pointing to your profile under /nix, and ~/.nix-profile/bin will contain the executables of what you installed with nix.
This isn't just a question of putting files under something other than /. pacman (and maybe apt, too) has a way of installing packages under different directories, but the files in the packages themselves will typically have references to the libraries, executables and other files they depend on assuming they've been installed under /. So, in order to use those installed packages, you'd have to chroot into the directory you've selected to act as root, so the references will work right. Nix, on the other hand, made sure to edit the references each package has to their dependencies so you can simply execute something in ~/.nix-profile/bin and each successive dependency will resolve its dependencies to build-specific directories under /nix.
You can use nix with pacman or apt-get or yum or any traditional package manager you'd like and they won't conflict. This is what is meant by being distribution agnostic.
So, if I will recompile whole distribution with other values for system directories variables, say /rpm/etc instead of /etc, /rpm/lib instead of /lib, etc., then rpm/yum/dnf/apt will be finally distribution agnostic in your eyes?
Nixpkgs is a whole collection of software (a distribution of sorts, but not a Linux distro), managed by Nix, which works identically on all Linux distros. It's also the package collection for NixOS.
Nix is cross-distro in the ordinary sense you describe, of course. It is also more deeply cross-distro in that:
• if you are trying to package for Nix in a normal way (the tools make it easy to do things this way and hard to do things otherwise by their nature), _all_ package collections for Nix will be cross-distro in this way
• The Nix community has a collection of thousands of cross-distro packages you can already use right now
`pkgsrc` is similar in this respect, and takes a more conventional approach, like the one you describe in this comment. I think that's a meaningful sense in which it is cross-distro.
Another way to think of this is that tools like `rpm` and `dpkg` or `dnf` and `apt` are portable between distros _as far as distro developers are concerned_, whereas package managers/package collections like Nixpkgs and pkgsrc meet the portability needs of users of _existing distros_.
