Uh, is the U.N. a government? Maybe it wants to be. But it isn’t.
The point of the clause is that the president shouldn’t have divided loyalties.
As president and UN chief he would have a conflict of interest. Should he do what is best for the USA or what is best for the UN?
It many not techically be a government, but in spirit the UN is. The have a military authority (that uses troops borrowed from various countries) and they have at times taken over the governing of countries that have fallen into chaos.
Most importantly, the UN has waged war. The Korean war was authorized by a UN vote. Technicall, the UN was in charge of the Korean war.