Solved my problem. The server started, appeared in the list of Steam servers and I logged into the server without any problems.
It turned out that you need to transfer the steamclient.so file, located along the path /home/steam/steamcmd/linux64/steamclient.so (for x64 OS), to the folder with the server files /home/steam/steamcmd/pz_my_server/projectzomboid.
pz_my_server - server root folder.
The answer was found here: https://steamcommunity.com/app/108600/discussions/1/1457328927845762140/