All religions have similarities in them. This is because God revealed elements of
truth to those who were seeking truth, before Christ came. When Christ came,
he revealed God as a Father, and who is full of mercy and love. It took off for four
centuries. However, once the Church leaders got involved with the Empire of the ruling governments, they lost the tenets Christ had taught. Over the centuries, various reformers were sent by God to reform Christianity. People like St Francis of Assis, and others.
Interior prayer, or contemplative prayer as we call it today, goes back the the 4th Century Fathers of the Desert. Abba Isaac and St John Casian, then of course the Carmelites in the 12th century. Contemplative prayer got lost through the centuries until the 1500's. St Teresa of Avila and St John of the Cross reformed the Carmelite order and founded the Discalced Carmelites. Their focus was union with Christ, and they saw that interior prayer was the way. However, from the Council of Trent on, contemplative prayer was pretty much kept in the monasteries and even when I was growing up in the Catholic Church before Vatican II, I had never heard of it.
Then, in the early 1970's, as many young Christians were going off to join Eastern Religions like Buddhism and Hinduism, Pope Paul VI, asked the leaders of the contemplative orders to being dialogue with the leaders of Eastern religions, to see what was drawing young people away from Christianity. The Trappists in Spencer MA invited some leaders of a nearby Buddhist meditation center to come to St Joseph's Abbey, and show them their method of prayer. As soon as the monks saw what was in Eastern meditation, they automatically knew that contemplative prayer was part of Christianity going back to the 4th Century. So, they thought of ways to teach young Catholics about contemplative prayer, in the Christian tradition. It originally was called "Quiet Prayer," but soon got the name, "Centering Prayer."
In all, it's Christ centered prayer. Just as mindfulness requires being centered on Christ. Spiritual growth requires that the person to be mindful of being in the presence of God at all times, and through contemplative prayer, to turn within where God dwells and be mindful of His presence. St Teresa of Avila called it "Mental Prayer."