"Christians and Muslims believe that Abraham was a Prophet,
Muslims and Jews consider the swine unclean,
Therefore, Christians consider the swine unclean", doesn't necessarily follow.
Also, to deny the existence of religion just because one cannot find its essential soul is to only affirm a paradox. For instance, take the argument for the denial of the car which says that the car really doesn't exist because when one begins to take apart the car, there will eventually come a point when the car ceases to exists. For instance, I begin by removing the tires and would still be capable of saying that the car exists, but doesn't have tires. Or say, I begin by removing the door and would still be capable of saying that the car exists, but without a door. However, as I begin to take away the parts of the car one by one, I finally realize that there comes a point when I cannot call the car a car anymore. However, does this mean that the term "car" is useless?
The above is an example of Sorites Paradox. The Sorites Paradox usually asks the question, "Suppose there is a heap of sand; if I remove a grain of sand the heap will still be a heap; if I remove another grain, it will still be a heap: how many grains must I remove from the heap in order for the heap to cease to remain a heap?"
The above is called a paradox because we know that the heap does exist; however, when one tries to define a heap with reference to specific number of grains, the definition becomes impossible and "heap" becomes nonsensical.
Somewhere, there comes a point when the abstract concepts cannot maintain themselves before the empirical concepts. But, the case can also be vice versa. Take Zeno's paradoxes, for instance.
However, one must not forget the historical question as well. For instance, Hindu was never considered an ism in early history. The ism was suffixed much later. "Hindu" didn't refer to a religion, but to a people. In fact, the ancients used the term to refer to the land. For instance, Esther 1:1 refers to India as hodu. Similarly, with regard to "Christianity", it was the disciples of Christ who were first called "Christians" in Antioch. But, noting that now a term such as "religion" has already become a part of common parlance, to yet avoid confusion and ambiguity, one can use more specific terms like, say, Vaishnavites and Pentecostals rather than Hindus and Christians. Not that we can't call them so; but, that it is important for a communicator to be clear and specific in communication....