This is really more a logic/philosophy question than a Christianity one.
And it is logically similar to this scenario:
Twin brothers get separated from their mother and from each other at a young age. As adults they later reunite and discuss their mother, to find that their childhood memories are imperfect, and sometimes conflict. Does this mean they have multiple mothers?
While the brothers may have different memories or opinions about their mother, they are still borne of the same mother.
I love my boss. My coworker hates the same boss. Does this mean we have multiple bosses?
Differing and conflicting perceptions and opinions of the same being/entity/object does not mean the being/entity/object in question is actually multiple beings/entities/objects. It simply means our perceptions and opinions are different and often times imperfect.
Following from these examples, it is clear that the twins, and me and my coworker do not indeed have multiple mothers/bosses. Likewise, Christianity does not indeed worship multiple Gods.
Going a step further, to more directly address the actual question, nor do the twins functionally have multiple different mothers--that is to say, one doesn't send Mother's day cards to one city, and the other to another city. Nor do they function as if they have multiple mothers; that is to say, neither brother sends Mother's day cards to multiple addresses.
Nor do my coworker and I functionally have multiple bosses; I don't call my boss on one extension, while my coworker uses another. Nor do we function as if we have multiple bosses; I don't call one of multiple extensions when I need a PO approved.
And as applied to Christianity, the various sects/denominations/faiths/traditions all functionally worship the same God1, so they do not functionally have multiple gods, and no Christian individual prays to multiple gods2, thus do not function as if they are polytheistic.
1Some minority Christian faith groups try to tie Christianity to other non-Christian belief systems; such as Christianity and New Age, where the nature of God is "changed" (i.e. "God is the Universe"). In such a belief system, where the adherents are worshiping "the Universe", they are not in the strictest sense worshiping the same God that main-line Christianity worships, who is distinct from the created Universe. In this regard, and in this regard only, it may be possible to claim that Christianity as a whole is "functionally polytheistic," in that two subsets worship each a distinct [Gg]od. But in my mind, this still is not the same as "functionally polytheistic."
2Some Christian sects do have polytheistic undertones. Mormonism for example, has some polytheistic (or henotheistic) undertones. Sometimes Catholicism is considered "polytheistic," especially in certain parts of the world where praying to the saints is hard to distinguish from worshiping the saints. But these possible examples of "functional polytheism" have nothing to do with the various Christian faiths, and have everything to do with the specific faiths being (or appearing to be) truly polytheistic, so I don't believe they apply to this question.
I think it comes down to a question of identity (or the external appearance of identity) of the [Gg]od(s) being worshiped.
Christian sects, with rare exception (see 1 above) believe in a God that share the same identity. They all believe God wrote (either directly or indirectly) the 66 books of the Bible, they believe God became human flesh in the person of Jesus Christ, and they believe God died a torturous death on behalf of sinners.
Whether a Christian group worships the God thusly identified through song, acts of service, acts of war, or any other means, the identity of the God is the same.
This is distinct from the question of whether Muslims and Christians worship (or appear to worship) the same God, because in this case the identity of the two Gods is distinct (at least on the surface). Christians believe in a God identified above. Muslims believe in a God that never took on human form, and who sent a final prophet, Muhammad, etc.
(Whether, upon further investigation, we can determine that the identity of the [Gg]od worshiped by Christians and Muslims is indeed the same is for another discussion. The only point I'm making here is that it's easy to see how they may appear to be different Gods.)
Based on this identity-of-God argument, I would say that Christianity as a whole (not withstanding fringe groups that subscribe to an alternate identity of god), is not functionally polytheistic.