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I use "artificial consciousness" as a broad term to describe the possibility that a computer may have the same experience of reality and of itself that we have.

I guess that a religious view, more precisely a view that requires humans to have a "soul", is fairly incompatible with the concept of "artificial consciousness".

What I am asking instead is why does such rejection sometimes come from a non-religious view?

Shouldn't a non-religious view lead almost immediately to the acceptance that humans are nothing more than machines themselves, so that every difference between humans and computers is merely architectural (biological neurons vs transistors)?

I hear sometimes people messing with the fact that the human brain has a particular structure that, mysteriously, cannot be reproduced through computation (violating the Church–Turing–Deutsch principle). This argument would require some super-natural properties related to the human brain, going back to a religious view that humans are "magical".

John R. Searle is a non-theist who believes in biological naturalism. Wikipedia describes Searle's position as:

Searle denies Cartesian dualism, the idea that the mind is a separate kind of substance to the body, as this contradicts our entire understanding of physics, and unlike Descartes, he does not bring God into the problem. Indeed, Searle denies any kind of dualism, the traditional alternative to monism, claiming the distinction is a mistake. He rejects the idea that because the mind is not objectively viewable, it does not fall under the rubric of physics.

If one has consciousness coming from a program running on a Turing machine, which is what I assume is meant by "artificial consciousness", one has dualism.

Searle expressed his concern against the dualism of strong AI in his paper, Minds, Brains and Programs, where he described the Chinese Room Argument:

This form of dualism is not the traditional Cartesian variety that claims there are two sorts of substances, but it is Cartesian in the sense that it insists that what is specifically mental about the mind has no intrinsic connection with the actual properties of the brain. This underlying dualism is masked from us by the fact that AI literature
contains frequent fulminations against "dualism'-; what the authors seem to be unaware of is that their position presupposes a strong version of dualism.

So, one reason for non-theists to reject artificial consciousness is because it implies a strong form of dualism. When one moves the program from machine to machine, if that program is indeed our minds, then we have gone through an out-of-body process to be reincarnated in another body.

Searle, J. R. (1980). Minds, brains, and programs. Behavioral and brain sciences, 3(3), 417-424.

Wikipedia contributors. (2019, February 5). Biological naturalism. In Wikipedia, The Free Encyclopedia. Retrieved 19:20, April 22, 2019, from https://en.wikipedia.org/w/index.php?title=Biological_naturalism&oldid=881932833