The question everyone asks about the Megillah is: Where is God? Now, on a certain level this question is improper. A fundamental Jewish belief is that God keeps a close eye on the world and interferes when necessary (even if this is not always obvious). To assume that God does not have a say in the natural world, that without a deus ex machina there cannot be a deus, is heresy (the "charedi" heresy, I might add, if I wanted to be political). And so even if there are no miracles and no speeches by God or a prophet, Megillat Esther could still be a completely religious story, with God simply the clear actor behind the scenes.
But while you could have a religious story without miracles, it's harder to say that you have a religious story without religion. And Megillat Esther is missing not only God's name, but almost any religious content whatsoever. To summarize the Megillah: The Persians have parties; one Persian decides to exterminate the Jews; two Jews get courageous and prevent the extermination; the Jews are safe. Where exactly is the religious content here? Why does the Megillah deserve to be part of the Bible more than Michael Oren's "Six Days of War"?
There have been at least two clear Jewish responses to this issue. The "midrashic" response focuses on the phrase "kiyemu vekiblu". Its simple meaning is that the Jews agreed to celebrate the new holiday of Purim for all generations. According to the midrash, though, it refers not to the holiday of Purim alone, but to the entire Torah. While at Mount Sinai the Jewish people accepted the Torah under the "coercion" of that unique set of circumstances, in the Megillah they accepted the Torah willingly,"kadat ein ones", as a response to the events of the time. Thus, historically, the Megillah story was the impetus to repentance and religious change. As such, it certainly merits being called a religious story. This interpretation is beautiful; its main deficiency is that it has only a limited basis in the text of the Megillah.
The "satirical" response, made explicit by recent Tanach commentators but with much earlier roots in Jewish tradition, says that the Megillah in fact does have a clear religious message. According to this theory, the Megillah includes pointed allusions to the land of Israel, Jerusalem, the Temple, and to certain prophetic books of the Bible. In addition to the "superficial", historical meaning of the story, there is also a clear polemical message: Why did you adopt Persian culture? Why didn't you return to Israel? Why do you regard Ahashverosh instead of God as "the" king? Presumably, though it takes an expert to notice this message today, it would have been immediately obvious to anyone in Mordechai and Esther's era.
I think that, without denigrating either of these responses, you can still hold that the "main" theme of the Megillah is national, not religious. The central story is about physical survival, and there is nothing wrong with that. The holiday of Purim is of course a mitzvah - but it may be primarily a mitzvah bein adam lechavero, a holiday of friendship and mutual solidarity, more than a ritual observance or mitzvah bein adam lemakom. Clearly, there is necessarily still some religious aspect to the holiday. The Jews were after all saved by God through a miracle - whether it was hidden or not is irrelevant. And of course, the continued existence and activity of the Jewish people by definition has religious significance. But at the same time, it seems there is a distinct aspect of the holiday which does not depend on religiousness. Hopefully, this aspect can form a basis for relating to the many Jews who, while not religious, nevertheless enthusiastically celebrate Purim and holidays like it such as Chanukah.