No, it doesn't. The Holy See is considered a "non-member observer state." The only entities with this status are the Holy See and the State of Palestine.Actually, the Baha'i Faith has the same status in the UN, and I believe the Baha'i Faith was the first to have 'consulting status' as of 1948.
There's also observer status, which non-state entities can get (e.g. the Red Cross and the European Union both have observer status), but AFAICT, no Baha'i organization has observer status.
This list from the UN says that the Baha'i International Community got special consultative status in 1970, not 1948.
Edit: consultative status is a significantly lower level of status than observer status: Observer status gives an organization privileges with the UN General Assembly, while consultative status only gives the organization status with the UN Economic and Social Council (ECOSOC). And (somewhat counter-intuitively) special consultative status is a lower level of status than general consultative status. Organizations with general consultative status are deemed by the UN to be concerned with most of ECOSOC's activities, while organizations with special consultative status are only concerned with only a few fields of ECOSOC's activities.