Hi Tom,
Naturally there's a confusion between ArcGIS for Desktop products family (hence "Classic" apps such as ArcMap and the new family member - ArcGIS Pro), to their Licensing method.
ArcGIS Pro is a Desktop app, but it's licensing is not software based (like "classic" Desktop apps - Single-use/Concurrent which can be permanent), but User identity based.
The Pro license is aquired by signing in to the organizational Portal (ArcGIS Online in your case) and not by a license server.
Further more, the Pro license is entitled by your Desktop Licenses under active maintenance agreement (usually renewed every year), so it has an expiration date driven by the maintenance agreement.
Anyway, I'm glad to hear that you got read of the expiration messages.
Regards,
Yovav Z.