I find it a bit challenging to think of it as a singular school considering that it encompasses both Christian philosophers like Kierkegaard and atheist philosophers like Sartre who take its premises in very different directions.
I don’t think the notion of a ‘school’ — as suggestive of common adherence to some more-or-less rigidly defined body of doctrine — is usually a particularly useful one in philosophy, though undoubtedly it has had an unfortunate degree of traction in the continental tradition in particular. If we instead think of existentialism as picking out a fairly recognisable, broad set of commonalities, then it is obvious why Kierkegaard and Sartre would both be included under that label — above all, their shared emphasis on the terror and arbitrariness of choice; where their religious differences come into play is more in their differing proposed responses to this same diagnosis of the human condition (and if we make the rather unoriginal observation that ‘humanism’ (which Sartre of course thought existentialism was a form of) often draws heavily on secularised aspects of Christian (especially Protestant) morality, then the religious differences can perhaps also start to seem somewhat less stark).
Of course you are right though that the term does have a very broad usage (at the limit, it can sometimes seem like it is just being used to mean all French and German philosophy from the middle of the 19th century to the middle of the 20th), and many thinkers that have been described as ‘existentialists’ have importantly disagreed with one another on a variety of matters.
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