Tosofos and Christianity
Most rishonim considered Christianity to be avodah zarah. A notable exception were the Tosafists. They felt that the trinity is shutfus (assigning partners to G-d). That the Father is the Creater, and identiable with the Jewish G-d, and the other persons of the trinity are minor deities that mediate. The practical distinction is that while Jews are prohibited from believing in shutfus, it is permitted to non-Jews under the covenant of Noah.
The Tosafists are a major force in Ashkenazic ruling. On their ruling, many Orthodox Jews who work in jewelery sell crosses and crucafixes. A necessary factor is the assumption that the overwhelming majority of customers will be people who aren't ethnically Jewish. Others do not rule like the Tosafists.
Another issue is whether the Tosafists' statement about the Catholicism of their day applies to any / some / most of the plurality of Christianities that exist today.
The status of Christianity in terms of the halachic category "avodah zarah" is unclear, as there is a dispute among both the rishonim and the acharonim as to whether Christianity is halachically allowed for non-Jews. But certainly most commentators, rishonim and acharonim alike, agree that Christianity is certainly avodah zarah for Jews, with the argument being only to non-Jews.
The Rambam ruled in three places that Christianity is idolatry and thus forbidden to gentiles. Mishneh Torah, Hilchos Ma'achalos Asuros 11:7 Mishneh Torah, Hilchos Avodas Kochavim 9:4; Peirush HaMishnayos?, Avodah Zarah 1:3 and implies the same in a fourth place (Hilchos Melachim 11:4) But most of these texts are missing from the the standard Vilna edition of these works.
Quote: Christians are idolaters and Sunday is their holy day (Mishneh Torah, Avodah Kochavim 9:4).
Quote: Know that this Christian nation, who advocates the messianic claim in all their various sects, all of them are idolaters. On all their various festivals it is forbidden for us to deal with them. And all Torah restrictions pertaining to idolaters pertain to them. . . . We deal with them as we would deal with any idolaters on their festival. (commentary on Mishnah, Avodah Zarah 1.3)
The Rambam also implied this in his Epistles to Yemen, but he also says Christianity has a role to play in G-d's plan by "preparing the way for the Messiah's coming and the improvement of the entire world..." (Mishneh Torah, Hilchos Melachim U'Milchamoteihem) However, in the Moreh Nevuchim (1:50) he writes that Christianity has a mistaken understanding of G-d, similar to those who assign attributes to Him. This would imply that Christianity is a min (deviant sect), which is perhaps a step up from idolatry.
Rabbenu Tam and his fellow Tosafists did not condemn Christianity as idolatry. "The fact that most people do business with the non-Jews on their holy days is problematic... It would seem that the reason for this permissiveness is that the non-Jews among whom we live are not to be considered idolaters..". Rabbenu Tam was of the opinion that the prohibition of doing business with idolaters before and during their holy days was only meant to apply to items that they might use in their worship, and did not apply to buying from them... (Tosafot to Avodah Zarah 2a).
When a certain rabbi in Europe prohibited all contact with Christians on their holy days (which was more often than not when the great fairs were held) Rabbenu Gershom objects, "But in the Land of Israel it is already customary to barter with non-Jews on their holy days, and we should not forbid this. It is better that the Jews contravene the law in ignorance than that they should do so knowingly, which they will inevitably do since their livelihood depends on their wares and most days of the year are Christian holy days."
The position of the Tosafists is complex. It is generally held that the Tosafists, particularly the Ri and possibly also Rabbeinu Tam, considered Christian belief to be "the partnering of the Name of Heaven with something else", and that as Noachides (i.e. non-Jews) are not forbidden to engage in such partnering; Christian belief and worship is permitted for non-Jews, and it is permitted for Jews to cause them to actively express that belief or engage in such worship. However, this judgement is not followed through in other areas. It seems from the words of Tosafot to Bechorot 2a and Sanhedrin 63b that their comments are limited to the case of oathtaking, i.e. partnering the Name of Heaven with something else while taking an oath. The halachic context would not be the prohibitions against idolatrous worship but rather the prohibition "Let them not be heard as a result of you", which would be understood as "let them not be heard as the exclusive guarantors of an oath as a result of you. Several great acharonim, including Shaar Efraim, Noda B'Yehudah, Meil Tzedakah, Olat Tamid, and Chazon Ish, understood Tosafot in this fashion. Some of them also interpreted Ramo, who cites Tosafot, the same way. But there are also those, including Shach and Seder Mishnah, who read Tosafot broadly, as referring to all prohibitions associated with Christian faith. But according to either reading of Tosafot, Jews holding Christian beliefs are violating avodah zarah prohibitions. Even Meiri, who assigns Noachides "bound by the ways of religion" (which presumptively includes Christians) a status equivalent to Jews for several legal purposes, probably believes that they are nonetheless worshipers of avodah zarah
Even if you conclude that Christianity is Avodah Zarah for non-Jews, it is not clear that most non-Jews are deeply attached to their avodah zarah beliefs, and further more there are Christian groups whose beliefs differ greatly in areas touching the halachic definition of Avodah Zarah. See Rav Henkin's citation in Bnei Banim 35 of Rabbeinu Yerucham's description of the non-Jews of his time as "not deeply attached to Avodah Zarah", and the Raavan spoke about this as well, and Rav Henkin sees in him grounds for distinguishing among the various churches.
On another matter, the Rambam was of the opinion that Islam is not idolatry. A certain Ovadyah converted from Islam to Judaism. As part of his learning his teacher told him that Moslems were idolaters. Ovadyah wrote to Rambam asking which of them was correct. Here is part of Rambam's responsum: "Your teacher reacted inappropriately when he caused you distress...Even if he had been right and you wrong he should have spoken to you politely and softly; how much more is this the case when it is you who was correct and he was the one who was mistaken"