If EasyJets systems are anything like their customer service, their in-flight food, their baggage handling or their scheduling, this is not surprising.
Getting compensated should be practically instant though, and definitely not take a year, so something went terribly wrong there.
Not really.
Even if you're legally entitled to compensation Easy Jet is famous to let you jump through so many hoops, lie to you, hang you out to dry, let you wait forever until you just give up.
Outsourcing to some company like Airhelp (which, charges 35% of recovered compensation if successful) may be a viable option to just save you the headache, while sticking it to the carrier trying to stiff you.
Not only did their system have me logged in another flight in another continent (!), they flat out denied I had a claim even though I had to stay put more than 24h, a good amount of that in the plane.
After getting sued, they tried to backhand contact me (not my lawyer) to pay compensation. Court made them pay.
This is all calculated.
I always found it interesting how they have much better tea than British Airways.
Generally you are right yes, it is airport service under one of the operating company of the airport.
However easyJet and other low cost airline have generally a very vertically integrated system where they try to operate almost everything themselves to reduce cost.
Some airport have entire dedicated terminal for them, I would not surprise if they manage also their luggage system in these airports.