How is this movie Existential?

The OP's question is actually a very good one. While Bergman is generally, and rightly, characterized as an Existentialist director, what is it about this particular film that makes it an example of Existentialist filmmaking?

In general, I think Existentialism is understood to be a set of philosophical understandings that recognizes that the essence of human experience lies in existence. By that is meant that the nature of man's existence, meaning the way man exists in the world, is the focus for any attempt to find meaning and purpose. This consideration must be not only the beginning point of all analysis of the search for meaning but as a frame of reference that must always be kept in mind.

I personally find Martin Heidegger to be the most helpful and coherent of Existentialist philosophers and theorists. To cut to the chase here, as it were, Heidegger posits that the most significant aspect of man's way of being is that it is being towards death. In other words our existence (at least on this world - he does not speak to, and certainly does not rule out, the possibility of an afterlife, whether as described by one of the world's religions or otherwise; in short this is not an atheistic philosopy, although of course atheists can also look on it as not contradicting an atheistic view) is most notable for being finite, existing in a time space continuum, with an eventual end. And we all know it.

In turn the recognition of this fundamental aspect of human existence brings with it a varying level of anxiety, felt as a varying concern as we face or turn away from this fundamental aspect in the course of our everyday lives. The anxiety in general terms reflects the difficulty we see in finding meaning in our lives when we know that our lives are finite. This in turn means that the turning away from recognition of our being towards death is also a general, in fact universal, aspect of human existence. But however much we turn away from such recognition, the true nature of existence is always there.

Given the foregoing, it should be obvious why The Seventh Seal is an Existentialist film. Not only is there the obvious interplay between Antonius and Death, and death for him as an impending and not removed event. There is also the ongoing discussion of Antonius's questions of what meaning he sees, in his life and in the lives of others, as he directly faces the prospect of his death. The backdrop of the Plague and the deaths that are resulting from it, all while people continue to try and lead lives with meaning, helps to focus the examination as well.

The specific religious references, I think, represent Bergman's then ongoing personal religious debate. Those familiar with his history know that he continued struggling with these more directly religious issues through Winter Light specifically, and more obliquely in The Silence, before beginning to continue addressing Existentialist themes, but without the overt focus on the theological, beginning with Persona. It is also of course generally true that Bergman eventually described himself as an atheist, and I would not dispute one could say that is where the examination of such themes as we see in The Seventh Seal eventually led him.

But I do not think it necessarily led him to that conclusion, if you will. And the film itself, despite its mostly dark feel and the pervasiveness of death within it, does not resolve itself by choosing atheism.

I hope that is helpful.