I image neutral systems as being members of the Republic with representation at the Senate, and the intent to stay in the Republic, unlike the Separatists. However, they think other members should be able to leave the Republic if they so wish, so that’s why they don’t support the war effort.
I’ve looked into this more - it’s complex but you’re right. The Galactic Republic, governed by the Galactic Senate, represents a republic of worlds. Neutral systems, including Mandalore, are OUTSIDE of that system - they are not members of the Republic. HOWEVER, there is a faction WITHIN the Galactic Republic called the Pacifist Faction, who have an additional seat WITHIN the Galactic Senate for the Council of Neutral systems. This seat exists deliberately to allow allies and neutral parties from outside the Republic to at least have a formal voice and presence within the Republic beuraucracy.
So Mandalore is not a member of the Republic, but as the leader of the Council of Neutral Systems is granted a seat in the senate.
Or to put it another way, “You are on this council, but we do not grant you the rank of member state. Take a seat.”