Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
MSFT only sells Office 2016/365 and Publisher 2016 now. You can buy a stand alone version of Publisher 2016 from MSFT however it may not install and run along with Office 2013. The two versions of Office do not play well together. Plus MSFT has hobbled Pub 2016 in a number of ways and I view it as inferior to Pub 2013 and older.
Reference: "We need to remove some older apps" error : https://support.office.com/en-us/article/-We-need-to-remove-some-older-apps-error-a225347f-e102-4ea6-9796-5d1ac5220c3b?ui=en-US&rs=en-US&ad=US - Office 2016 (whether part of an Office 365 subscription or purchased as a suite of applications such as Office Professional 2016, Office Home and Student 2016, or Office Home and Business 2016), cannot coexist on the same computer as standalone Office 2013 applications.
I would suggest that you Google for an stand alone copy of Publisher 2013 or better yet Pub 2010 which is the best version of the three IMHO. And personally I prefer Pub 2007. Pub 2013, 2010 and 2007 and even older stand alone versions will install and run just fine beside your version of Office 2013. Just make sure you always do a custom install and do not overwrite any other versions that might be on your computer. Reference: https://support.office.com/en-us/article/install-and-use-different-versions-of-office-on-the-same-pc-6ebb44ce-18a3-43f9-a187-b78c513788bf
Depending upon what you are going to use it for, you can even install and run Pub 2003 and 2000 on a Win10 machine.
DavidF