> That shouldn't be a uniqueness constraint as a country may have multiple capitals
And, in theory, two countries could share the same capital (as a condominium which is equal joint territory of both, or some kind of third territory belonging to neither), although I'm not aware that's ever actually happened. But, at the subnational level, it has happened in India – Chandigarh is the joint capital of two Indian states (Punjab and Chandigarh), each of which has it as its capital despite it being in neither of them, as well as being a union territory (and hence also capital of itself).
And, in theory, two countries could share the same capital (as a condominium which is equal joint territory of both, or some kind of third territory belonging to neither), although I'm not aware that's ever actually happened. But, at the subnational level, it has happened in India – Chandigarh is the joint capital of two Indian states (Punjab and Chandigarh), each of which has it as its capital despite it being in neither of them, as well as being a union territory (and hence also capital of itself).