Religion without the existential question is not really religion. It sinks into something external and almost trivial; almost like a performance in which we are religious actors. We do various observances and ceremonies. We learn to hide in uniformity and conformity.
The existential question concerns first-person knowledge and experience, for example, who am I (essentially)? Or it can be something like, what is the meaning of life (its essence)? or what happens when I die?
The existential question starts as a certain kind of thought about oneself that turns into a special kind of question. In this, I am thinking about myself as myself. I am even trying to look beyond my thinking into sheer no-thought where the subvocal “myself” is still unmanifest.
So, can Zen Buddhism answer the existential question making it a religion? The answer is yes insofar as it provides the necessary answer to various kinds of existential questions when we have seen our true nature (kensho). At the least, Zen Buddhism opens up an existential dialogue so that we may come to realize ourself or at least get oriented to the proper direction of inquiry: a looking within you could say.