Yes, there is the common security and defence policy, and the Article 42 of Lisbon and all that, but it all still relies on national systems.
> 18 April 1951 – European Coal and Steel Community
> Based on the Schuman plan, six countries sign a treaty to run their coal and steel industries under a common management. In this way, no single country can make the weapons of war to turn against others, as in the past. The six are Germany, France, Italy, the Netherlands, Belgium and Luxembourg. The European Coal and Steel Community comes into being in 1952.
https://european-union.europa.eu/principles-countries-histor...
Why wouldn’t a unified permit to prove you registered for mobilization be relevant to what the EU is for?
Defence and the military is a sovereign matter that has nothing to do with the EU... except we are seeing that this is changing without democratic national mandates.
This is the EU describing its own history and beginnings.
I can only repeat that defence is a sovereign matter in which the EU has no power, but there is a trend of changing this by making it happen as "fait accompli", especially since the war in Ukraine, which is used as pretext.
There is a new military Schengen project to make troops and unified military documentation across whole EU.
Obviously there will need to be a registry of personnel there, so these people can be prevented to leave.
On the side you have SIS Schengen, where you can (already) have an active arrest warrant for desertion.
Nothing indicates that European Union is going to fight against such registries. It's even the opposite.