While it's correct to note that for a very long time Eastern religions have discussed rejecting the material world, I don't think it's correct to say that it isn't Christianity "specifically" that discusses denying the flesh. In other words, though Eastern religions may also preach the denial of the flesh, it's nevertheless the case that Christianity specifically, if not singularly, discusses denial of the flesh.
This is irrelevent. The concept of denying the flesh is still there, and in many eastern religions, the whole point is to deny the material world. A very strong focus on self control and not giving into your base desires, which I would say is even stronger than Christianity's denial of the flesh, as in Christianity pleasure itself is not an issue, but extreme forms of it is.
Christianity was tempted by the same error that lives and abides in the Eastern denial of the flesh. That temptation as it presented itself to Christianity was called "Gnosticism." Gnosticism was an early attempt to bushwhack Christianity into the same error as Buddhism: the complete demonization of the flesh. Christianity overcame Gnosticism and for that reason became the seedbed for the modern technological revolution that is transforming the terrestrial and biological world into something far more permanent and spiritual than flesh and blood.
Eastern religion fell prey to Gnosticism and atheism. Eastern religion is neither theistic, nor willing and able to endure the flesh for the purpose of erecting the new body required by the spirit.
The soul concept is an ancient one. In fact it is so old that we do not know how it originated. So it comes as no surprise that Dawkins would refer to something similar, as unrelated cultures around the world came up with the concept independent from each other.
I still don't see how this specifically is taken from Christianity unless you are talking about cultural influence, as I see it as Dawkins figuring out what ancient religions have been saying all along.
Christianity teaches the concept of the trinity: body, soul, and spirit. In this conceptualism, the body is genes and biology, the soul is the self-consciousness/god-consciousness (they come packaged together) related to the cerebral cortex coming online for the first time.
In the concept of the trinity, the body evolves until the soul emanates, or emerges, once the cerebral cortex is developed. Then, when the cerebral cortex comes online, the individual achieves self-consciousness, i.e., the soul, which immediately allows god-consciousness as a product of self-consciousness.
Where Richard Dawkins, Jeff Hawkins, and Daniel Dennett (to name just a few famous atheist thinkers) come into the picture is when they use the self-consciousness derived through the cerebral cortex, to deny god-consciousness, for the sake of elevating the reptile-brain, and its world, to a form of conceptualism devoid of God: NeoDarwinism.
The early reptile-brain, the mammalian brain without the cerebral cortex, doesn't really have a conscious understanding of God, life, death, or the universe. Privileging the reptile or mammal brain over God, these atheist know of life, and death, through the cerebral cortex, but deny God.
They use the very organ given as the greatest gift of the spirit of God, through which he can be known and conceived (can literally enter into the flesh), but chalk its existence up to the processes in the flawed animal understanding that supposes the world is a real "environment" able to cause the so-called natural selection of the mutations in biology that arrive at something like the cerebral cortex. Only those who reject God at the point of god-consciousness could possibly believe something as banal as the idea that environments even exist apart from living organisms, and that they somehow select the right mutations to arrive at the cerebral cortex.
What we have meant to say is that all our empirical experience is nothing but the representation of appearance; that the things which we experience are not in themselves what we experience them as being, nor their relations so constituted in themselves as they appear to us, and that if the subject, or even only the subjective constitution of the senses in general, be removed, the whole constitution and all the relations of objects in space and time, nay space and time themselves, would vanish. As appearances, they cannot exist in themselves, but only in us. What objects may be in themselves, and apart from all this receptivity of our sensibility, remains completely unknown to us. We know nothing but our mode of perceiving them - a mode which is peculiar to us, and not necessarily shared in by every being, though, certainly, by every human being. With this alone have we any concern.
Immanuel Kant.
John