I don't know about the law of Anhalt in the 1890s, but, in any event under modern English law, non-commusation does not mean there is no marriage (as is the case with a bigamous marriage), it simply makes it voidable, i.e. one party can obtain an annulment if they wish.
Now, the 1890s is an interesting "End of Times" era in Anhaltinian law. On the 1st of January 1900 the new
Bürgerliches Gesetzbuch, Civil Law Code, came into force in the whole German Empire and replaced all the older territorial law codes. Which in backwards Anhalt (and the Thuringian Duchies!) actually was the
Sachsenspiegel, Saxon Mirror, from the early 13th century (augmentated by later decrees)! As far as I can see the
Bürgerliches Gesetzbuch, which still is in force in Germany, takes a rather liberal, Protestant approach to marriage and divorce and does not busy itself with the archaïc issue of consumation versus non-consumation and only opens for annulments in extraordinary cases (like incest and bigamy). Very simplified it says that if one or both of the marriage partners can't stand each other and don't want to live together, they can apply for divorce. But it's harder to find out what the
Sachsenspiegel, which Marie Louise and Aribert were subject to, has to say about the matter. Partly because it's written in medieval Low German!
Since Aribert's father dissolved the marriage as
summus episcopus, if I remember correctly, ecclesiastical law might have played a part. Article 6 of the House Law of Anhalt(-Köthen) of 1811 does give the ruling Duke authority to dissolve, ipso jure outside a court of law, all marriages contracted by members of his house without his consent, By analogy it might also have come into play, although the house laws say nothing about consumation of marriage and divorce.
Fyodor Petrovich Byergov, State Councillor to His Majesty the Emperor of All RussiaFriedrich Peter von Bergow, herzoglich Anhaltischer Justitzrath