Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
I suppose yes. Are you trying to install it from the 2nd disc?
I suggest you to insert 1st disc and run customized installation (uncheck all the components) using the product key and the try installing Publisher from the 2nd disc which shouldn't prompt you for the product key.
You could also try changing the installation folder, just to make sure that there is no conflict with Office 2013.