Why do Isolation Forest implementations turn it into a supervised learning problem (with random values for the target?)
Extra-random Trees needs a target variable, so Isolation Forest generates a random target (sklearn, solitude).  At prediction time, no y values are used, and the ExtraTrees doesn't actually make a prediction; instead, the samples are propagated to the leaves and the depth is extracted (sklearn).
As for the tree-building process, sklearn at least doesn't make use of the y values, because the ExtraTrees model has max_features=1 and splitter='random' (source).  I'm not so sure about solitude, since it has mtry=ncol-1 (source); maybe further down, using splitrule='extratrees' takes care of that?  Otherwise, the splits chosen will try to optimize on the random y, though since those are random it maybe doesn't matter (certainly I wouldn't call it a supervised model, anyway).