What does "psql: /usr/pgsql-11/lib/libpq.so.5: no version information available (required by psql)" actually mean?

You have installed a mixture of packets: some from the Fedora distribution, some from the PostgreSQL Global Development group.

These don't work well together. In particular, you have psql from the Fedora packages in the PATH, but libpq from the PGDG packages is on the library path.

That explains your question #2.

Ad question #1: that warning is printed if the library version of the installed library is lower than the one the software was linked with.

Remove the Fedora packages to get rid of the problem:

dnf remove postgresql postgresql-server

I hope that puts the correct psql on your PATH; if not, reinstall the postgresql11 package.

After experimenting and trying out suggestions made by @Laurenz Albe I've managed to reach solution.

Solution after performing mentioned steps in my question to do one more additional step which is to run sudo dnf reinstall pacakage_name (package_name is one of three below mentioned packages ) command for following packages:

[nikola@192 ~] $ rpm -qa|grep postgres
postgresql11-11.8-1PGDG.f30.x86_64
postgresql11-libs-11.8-1PGDG.f30.x86_64
postgresql11-server-11.8-1PGDG.f30.x86_64

After that everything works as it should.