Regardless of the Christian opinions about the holiness of books, I think for a discussion of Avodah Zarah we have to focus on practice. I'd focus on areas of practice: prayer, non-prayer tasks, theological questions, and the practice of engaging the Torah.
tl;dr - Christian Avodah Zarah is unique in it's claiming to be inheritor of Judaism, but rejects MOST of what Islam does still adhere to: Idolatry, Diet/Cleanliness/Practices, Theologically unique in understanding human sin, and finally, unique in approach to Torah.
the too long:
Christian prayer practice by and large is obvious idolatry. A figure or memory of a man/god is worshiped. That's pretty serious Avodah Zarah. Is it unique too Christianity? No.
Non-prayer tasks, such as diet, cleanliness, charity, etc are very different from Judaism. Most importantly, they consider their religion to be a nullification, or some strange sense of "completion" of the very laws we live by. I'd say that makes doing christian non-prayer practices very uniquely non-Jewish, as they're seen as the precise nullification of them. The willful nullifying and overcoming or superceding of Jewish normative life is the uniqueness to me. What KIND of Avodah Zarah would it be? It was be a conscious rejection, with no appeal to ignorance. It would be a kind of Avodah Zarah directly purposed to be destructive of Jewish life. How many other types of Avodah Zarah are so consciously and arrogantly violent to Jewish law and normative practices? Sharia? I don't know.
Theologically, what Christians believe - that there is a unique incarnation of the divine spirit that existed in order to setup and resolve an "original sin" and that human sin can ONLY be absolved through this incarnation of the divine spirit. This flies in the face of all Jewish thought - we know that the soul is pure, the soul is good, and that humanity has teshuvah available always to clear away the problems and reveal the inherent and ineluctable goodness of the soul. In this way, by the foundational concept that the human soul is somehow impure, this type of Avodah Zarah is directly in conflict with Jewish ideas. It creates a psychological state of being that is exclusionary and rejects other traditions. In this way, the Avodah of HaShem, seeing and living universal consciousness bounded at the very least by the seven laws of Noah, is made impossible. Know this, as christians who know Judaism must, they are again arrogantly trying to replace fundamental Jewish theological ideas. It's the Avodah of Negation, perhaps, to coin a phrase.
Finally, wrt engaging Torah. Christians do not read the Torah according to our tradition. They read the Torah from a completely different point of view, without any reference to pre or post exhilic Jewish commentators, traditions, etc. They do not involve themselves in Mishnah, Gamara, or any of the ways we practice understanding Torah. As reading Torah is a normative Jewish spiritual act, an Avodat HaShem, they are espeically reading it in a way the cancels the Jewish tradition. Since they claim to be inheritors and supplantors of the tradition, I'd also call it a willful Avodah Zarah. This time I'd call it one of perversion of the mean, spirit, and text of the Torah, in order to teach falsehoods about the Torah.
So, to wrap-up, this Avodah Zarah is special, as it claims to be the supplantor of the Jewish tradition, it is often quite obviously idolatry despite their protestations, it casts out just about all the practical acts, warps the theological teachings to create a pitiful humanity imprisoned in sin, and willfully rereads the Torah to suit their theology, casting off the tradition from which the Torah came into the world and teaching from the "same" book the destruction of the Jewish people.
So, it's unique Avodah Zarah, indeed.