What was his status with regards to his Judaism?
There is an opinion (Rit Algazi, in his commentary to Ramban's Hilchos Bechoros 8:65) that the rule about the son of a Jewish mother and non-Jewish father being Jewish comes with a caveat: it depends on how he was raised. If (as was often the case in earlier times) a non-Jew impregnated a Jewish woman (whether consensually or not) and afterwards wasn't involved in his upbringing, so that the mother raised him as a Jew among Jews - then indeed that's what he is. On the other hand, if the father has custody of the child and raises him as in a non-Jewish environment, then retroactively he's considered to not have been born a Jew.
(It must be stressed that this is a solitary opinion. For practical halachah, CYLOR.)
Based on this, Beis Yitzchak (Even Haezer 1:29:8) suggests that Darius may indeed not have been considered halachically Jewish, since he was raised as Achashverosh's heir.
Another possibility is that he might be considered an apostate Jew, who for certain purposes is treated as a gentile. This argument is advanced by R. Yaakov Chagiz, in his responsa Halachos Ketanos, part 2, no. 240. (Daf Al Daf to Yevamos 23a cites this source, although they seem to understand him as referring more broadly to any child of a Jewish woman and a non-Jewish man.)
"The Ben Yehodaya cites the Zohar and the Arizal, that in fact Esther herself did not have relations with Achashverosh, but instead there was a "Shidah" - a demon - who appeared instead of Esther whenever Achashverosh wanted to have relations with her."
So according to this opinion Darius was not actually her son, and therefor was obviously not Jewish.
(Presumably a Shidah is a Succubus.)
Rabbi Hershel Schachter often discusses Darius with regards to the current question of religious Zionism. A "Jewish king"'s years are counted differently than a "non-Jewish" king. Darius was born to a Jewish mother but was king of Persia. Hence, in his younger years when he was more sympathetic to Jewish causes, that was enough for his chronology to work as a "Jewish king." Later in life he "soured" and thus while still legally Jewish as an individual, no longer had the conventions of a "Jewish king."
(Rabbi Schachter continues: if a Jewish-born king of Persia sympathetic to Jewish causes is enough to be called a "Jewish king", then certainly a state that identifies as Jewish -- complex as its relationship with religion may be -- qualifies as "Jewish government." Rabbi Schachter reads the mitzva of "establish yourself a king" as "establish yourself a Jewish government.")
Your real question here is weather or not the Midrash which says that Darius was the son of Queen Ester and Achavarosh is a true midrash, or if it's just there to teach us a lesson.
Darius was not Jewish, since he was Persian. There is no indication, outside of the medrash that he would have been Jewish.
This leads to two possible answers.
Either, The Midrash is not literal, or in the times of Darius, Jewishness was not passed on by the mother, but rather by the father.
Edit: It seems that the last line of my answer was not clear enough. I was not arguing that Jewishness is passed on by the father, I'm saying that if the Midrash is taken litterally, and Esther was his mother, but Darius was not Jewish, then that would mean that Judaism is only passed on by the father.. which is not the case.