Does the relationship-not-a-religion idea exist outside of American evangelical culture?

In a sense they are right though I am not sure of the context they meant. I have always understood the Bible and the writings of the Holy Fathers to indicate that Christianity is a way of life and not a religion. The late Father Alexander Schmemann wrote that Christianity was meant to be the end of religion because religion is only needed when there is a wall of separation between God and man. Christ because He is both God and man has broken down that wall and ended the need for religion. He also pointed out that religion tends also to only really deals only with matters of the soul and that Christianity is much more than just something to nurture the development of our souls. It is about mind, body and soul; our life lived in its entirety, all that we say, think and do being devoted and connected to God. And it is through our heart that we stay connected to God. But this connection is only understood and deepened if we devote everything we are and do to God and live a life of worship in which Christ-likeness is our goal. Hope this is helpful.