I understand what Christians mean when they say this. They are trying to express the idea that Christianity is different from all other religions— that Christianity isn't about doing a certain list of things or accomplishing so many goals. Christians often look at the Pharisees and call them "religious". Make no mistake about it, the Pharisees were definitely religious people. Christians, however, are also religious people (but in a different kind of way).
Take a look at this passage from James:
If anyone thinks he is religious and does not bridle his tongue but deceives his heart, this person's religion is worthless. Religion that is pure and undefiled before God, the Father, is this: to visit orphans and widows in their affliction, and to keep oneself unstained from the world. (James 1:26-27)Clearly, in this passage, religion is not a bad thing. James is basically saying that there is a lot of false religion in the world, but true religion is this: caring for orphans and widows and keeping oneself unstained from the world.
The word "religion," therefore, can be used in a negative way and in a positive way. There are many false religions, but there is one true religion. There are many man-made religions, but only one God-made religion.
It doesn't make sense for Christians to say that they don't have a religion, but a relationship. Why? Because anyone who isn't a Christian clearly recognizes the fact that Christianity is a religion— and they're right!
Some Christians seem to want to avoid using the word "religion" because of the bad connotation it has in today's culture. This argument, however, has some problems.
The word "Bible" also has a bad connotation in today's culture, should we avoid using it as well?
If you think about it, there are many Christian terms that have terrible connotations in today's culture, such as "holy" and "righteous".
It would be far better for Christians to abandon the phrase, "Christianity is not a religion," and instead explain what makes their religion unique. It would be better for Christians to say that they're the only ones who live out true religion. While it may be true that man-made religion is a cold, dead system of works, God-made religion is completely different.