The societies of the Americas and Oceania in the pre-European contact era had numerous different religions, nearly a unique belief system to each individual tribe. These belief systems had existed for hundreds, if not thousands, of years—generations of believers linking the present to the distant past. The culture shock of even the initial contact must have been extreme; a huge and abrupt paradigm shift which immediately challenged age old traditions.
While I am not sure of the current scholarship on this, I’m sure we all learned in high school that when Hernan Cortes initially made contact with the Aztecs they believed that he was the god Quetzalcoatl. After centuries of belief, their god had returned. And then millions of them died from diseases and bullets that Cortes brought with him. Needless to say, the psyche of the Aztecs must have been thoroughly destroyed.
While belief-destruction to this sudden degree did not occur everywhere, the traditional religions everywhere across the Americas and Oceania were systematically replaced by Christianity, most commonly Catholicism. Since the Jesuit society played a prominent role in the early history of many Spanish and Portuguese possessions overseas, the order instituted education systems which eliminated traditional beliefs and instill Catholicism in youth of the native tribes. Additionally, the intermarrying of Peninsulares and the native population further weeded out adherents to the old religions.
Religion is and always has been a very prominent identity marker across the various cultures of the world. With their religions destroyed, the natives of America and Oceania lost a primary pillar of who they were. Along with the economic and political changes which forced them into servitude and the violence which drastically reduced their numbers, native culture never stood a chance.