EU is indeed a military alliance. They are just no integrated command structures but this is common among most military alliances across the world.
SAFE credits are an EU military thing.
OK, the Treaty of the EU does have a mutual defence clause but the military is specifically out of scope.
Considering the huge ramifications of EU membership in all areas it still does seem ridiculously onerous if you are "just" after security and NATO-like alliances.
The mutual assistance clause doesn't oblige member states to support each other with military force even in case of aggression. A member can send blankets, food rations, or even just express diplomatic solidarity. Stretching definitions enough one can call it military alliance, but it is not in any practical form.