Microsoft Project Server 2013 will reach end of support on 11th April 2023. The Project 2013 desktop app also has the same end of support date.
If you have users of Project Server 2013 and your licenses are covered by current Software Assurance then you are entitled to upgrade to the latest on-premises 2019 release.
For a later release after the 2019 perpetual edition, Project Server is now available via a subscription which allows for on-premise use.
Project client compatibility:- You can connect to Project Server Subscription Edition with not only Project Professional 2021 and the Project Online Desktop Client but also with Project Professional 2019.
However, Project Server Subscription Edition requires Sharepoint Server Enterprise Edition Subscription and you can only upgrade from Project Server 2016 or 2019.
You might find the following links useful:-
Project Portfolio and Enterprise Resource Management (microsoft.com)
Software requirements for Project Server Subscription Edition – Project Server | Microsoft Learn
If you need independent advice with your Microsoft Licensing, Software Assurance & Subscriptions then please contact The SAM Club at [email protected]