The Western Interest in Buddhism seemed to start off with the theosophical movement, which seemed to be trying to prove that there was a common ancestry and essence of all religions.
Many western Buddhists are Buddhists, in a way that Christians would not define themselves as Christians. The Christians would probably define themselves as being part of a particular Church or sect rather than being "Christian". But perhaps most westerners come to "Buddhism" first and then take up a particular tradition later, though I suspect that this is changing. This means that they will mix practices from different parts of Buddhism within their own beliefs and practices.
Some Buddhist schools that have become established in the west have adapted their practices to make them more acceptable to the western way of life. This may just be the varying of the calendar of festivals to fir the working week. Or they may adopt forms and practices from other religions. The Order of Buddhist Contemplatives has translated the Zen scriptures and set them to English (Christian) plainsong. Zen Master Hogen Yamahata includes (Hindu) Hatha Yoga in his teachings, others may use (Daoist) T'ai Chih as forms of meditative exercise.
Christians have adopted Buddhist forms of meditation. The Jesuit priest Hugo Enomiya-Lasalle was a recognised Zen Master, and wrote books on Zen. There are similarities between Zen and Quakerism :- "Zen Buddhism holds quite a fascination for Quakers. It is iconoclastic, after the manner of early Friends, and seeks direct spiritual experience. At the same time, it claims to be nothing special, and sees the spiritual life and everyday life as being one. The practice of Zen has tended to become formalised, but its essence consists of 'just sitting' and reflecting upon our basic Buddha Nature. (Another name for the Inward Light.) There are Friends both in Europe and the USA who also practice formal Zen meditation, and find that this daily practice complements the meeting for worship" [page 70 of Pim, J 1999 - Listening to the Light - how to bring Quaker simplicity and integrity into our lives. Rider isbn 0712670203]
Some Buddhist writers (for example Rev Master Jiyu Kennett and Thich Nhat Hahn) have use the word "God" to describe Buddha Nature - in a way that makes me look at Christianity in a whole different light. Some (for example Thich Nhat Hahn and the Dalai Lama) have written books either looking at Jesus as a Bodhisattva or commenting on the Christian Gospels.
And writers of other religions have written commentaries on Buddhist Scriptures, For example Bagwan Shree Rajneesh has published books about The Heart Sutra and the oxherding pictures, as well as other Zen stories.
Is this the start of a new Theosophy? Where we accept the differences in the different paths of religions but find a similarity in the goal?
Well, perhaps not all of us - this message appeared on the Buddhism Depot discussion board (anonymously) in February 2000 - 'Hell Awaits Buddhists .... Jesus said "I am the Way, the Truth, and the Life. NO ONE can come unto the Father except by Me." "Neither is there salvation in anyone else, for there is no other name given under heaven among men whereby we must be saved." Hell awaits all of you, walking in your ignorance and your pride. You are sons of the Devil. You will burn and shriek hideously in a hell of fire FOREVER. Repent and believe the Gospel ! '