Suppose I write some package manager which will install, manage (start, stop, send commands, write configuration files etc.) binary programs distributed under various GPL licenses but not incorporate in my distribution. Can I license package manager under my own license of choice?
You propose to write a piece of glueware which on one end is designed to interface with a variety of standard free software publishing sites (github, gitlab, sourceforge, etc.) and download software from them, internally refactor it, then present it on the other end to the user's standard system software installer (apt, yum, dnf, etc.). You want to know if the licensing of your glueware will be affected by the software it installs.
Basically, no, I believe it will not. Taking a Red Hat distro as an example,
yum is licensed under GPLv2, but is used to install software with many licenses, including software under other versions of the GPL, under non-copyleft free licences, and proprietary software. More generally, the licence of a program doesn't bleed to either its inputs or outputs, so one could similarly have a proprietary installer which installed free software. As long as you are not conveying free software, but merely providing code which allows the end-user to convey it to him or herself, then I don't immediately anticipate problems.