Brew _is_ a linux package manager.

There is also conda/mamba/pixi/etc. (anything in the conda-forge ecosystem) that can be used without root. Then there are Guix and nix, which (mostly) require to be set up by someone with root privileges, but which then allow unprivileged users to install packages for themselves. I think I have even used emerge rootless-ly at some point a few years ago.