It's amazing how offended members of heretical groups will usually get when someone says their particular religion is not Christian. To be clear, there are true Christians in every denomination, so it's vital to make a distinction between the religion and the individual believer, but it's clear from the Scriptures that where fundamental divisions exist, either one side of the division is not Christian, or both aren't.
Let's use Mormonism as the example. It's interesting that Mormons should get so defensive about being called non-Christians, since the origins story of Mormonism includes God the Father telling Joseph Smith that Christianity had been corrupted shortly after the deaths of the Apostles (save John, who never died). In other words, Mormons don't (or at the very least shouldn't) accept Christians as just another equally valid denomination either, and originally, they did not call themselves Christian. But in my experience, Mormons generally get offended if you say Mormonism is not a Christian religion.
But if we're both Christians, why do we speak different messages? It's one thing to have opposing beliefs on minor issues. For example, infant baptism, yay or nay? Such a disagreement cannot be considered fundamental, because it says nothing about the way we are saved, or the nature or character of God. But the differences between Christianity and Mormonism are fundamental. The beliefs we have in common are overshadowed by the major disagreements we have in other areas.
As an example, Christians believe God is eternal. He never had a beginning, will never have an end (Psalm 90:2), never changes (Hebrews 13:8), there were no gods before Him or after Him (Isaiah 43:10) and God is a Trinity (Matthew 28:19). By contrast, Mormons believe "as man is, God once was. As God is, man can become." They agree that God has no end, but they believe He has a beginning. They believe He was a man that once inhabited another planet, serving under another god, and by serving that god diligently, he obtained godhood, which we also can obtain by obeying Him diligently. They also believe that Jesus is not God, but is rather the eldest of God's spirit children, and while I'm not aware if this is an official teaching, every missionary I've ever met has believed the Holy Spirit is not a conscious being, but is a force.
So there are a few fundamental differences between Christianity and Mormonism with regard to who God is, therefore we cannot be worshiping the same God. The Mormon Gospel is also fundamentally different. While Christians are saved by grace alone through faith alone, not of works (Ephesians 2:8-9), which are filthy rags (Isaiah 64:6), Mormons believe we are saved by grace "after all we can do" (Nephi 25:23), which Christians would say is not grace at all (Romans 11:6).
So, Christians and Mormons not only believe in a fundamentally different God, we believe a fundamentally different gospel. Aside from Paul imploring us to all preach the same things, because Christ Himself is not divided (1 Corinthians 1:10-13), he also says that there can be only one Gospel, and anyone preaching a different gospel ought to be cconsidered anathema (cursed by God). Therefore, if Christians and Mormons are preaching opposing gospels, no matter which way you cut the cake, at least one of us is wrong, or lying. Given that Mormonism is the religion that constantly runs counter to the Scriptures, it makes sense that they should accept the non-Christian label. If they want to be called Christians, getting offended isn't the answer, becoming a Christian is.
Lest anyone jump on the apparent contradiction of me saying a distinction should be drawn between a religion and a believer while simultaneously declaring if Mormons want to be called Christians, they should become Christians, I wish to clarify that I know first hand that not everyone who identifies with a specific religion actually follows it. I know two people who once identified as Mormons who took the name without taking the faith. The first is a young girl we'll call Paige for anonymity's sake. Paige joined the Mormon Church after developing a crush on a missionary. She called herself a Mormon and got offended if anyone dared question Mormonism, but was a Mormon in name only. She is impulsive, bouncing between religions according to what appeals to her at the time.
On the flip side, my friend, whom we'll call Ben, was also a Mormon in name only. He was, in effect, a sheep in wolf's clothing. Though a Mormon in name, attending a Mormon church, his Christian faith prevented him from swallowing the more absurd doctrines of the Mormon religion. It would have been unfair to call him a non-Christian, but at the same time, it would be odd to call him Mormon.
The distinction between religion and believer remains, then. However, when believer is indistinguishable from the religion, it becomes fair to equate them with it. Thus, if the religion is not Christian, and the believer is a true member of it, it becomes fair to also reject their claims to be Christian.
Though I have used Mormonism as my example, this principle applies to all heretical groups, and even to denominations which otherwise have no critical flaws, yet still preach disputable matters as infallible doctrine. In this case, it is morally wrong to say people in these denominations are not Christians, but it is fair to say they ought to preach the same things as other Christians, not cause divisions. The divisions they do cause, when contrary to Scripture, are still not Christian.
Why, then, is it so offensive to say so? In part, I blame our Relativistic society, in which "tolerance" is synonymous with silence, if not outright agreement. But it may also be considered conviction. Christ makes it very clear; you are for Him, or you are against Him. If you try to have both, you will be offended when you're told you can't. The question you must ask yourself is which you want more: To be for Christ, or against? If you are foolish enough to say "against", then do not be offended when someone says you aren't Christian; own that label all the way to Hell. But if you are wise, you will believe unto the saving of the soul, and will seek to reject that which He has rejected, obeying His every word, for as He says, "If anyone loves Me, he will keep My word; and My Father will love him, and We will come to him and make Our home with him. He who does not love Me does not keep My words; and the word which you hear is not Mine but the Father’s who sent Me." (John 14:23-24).