First of all, the Holy See (The actual name of the sovereign country we call the Vatican) is not a member state of the United Nations but a Permanent observer. This meant, until 2004, that the Holy See was not allowed to do anything except observe the deliberations of UN meetings. This changed in 2004 when it received the rights to:

...participate in the general debate of the General Assembly and to intervene in the discussion of any issue inscribed in the agenda of that assembly. It has the right to participate in all meetings open to all Member States, the right to make points of order and to exercise the right of reply, the right to circulate proposals and position papers as official documents, and the right to co-sponsor draft resolutions and decisions.

The reason of member state denial is not that it is a religion but that the tiny area of land it has and the small population makes it impossible to participate in many of the requirements of the United Nations. The Holy See is a sovereign nation and as such is not in the UN as a religion.

There have always been issues with the presence of the Holy See within the UN and protest continue to this day. Officially the Holy See is not the representative of Christianity/Catholicism but of the tiny speck of land in Rome. As such a representative of other religions is not the issue at hand here.