I think that it's fine to post the same question (or very similar questions) to Mi Yodeya and another SE site, as long as:
The two posts are on-topic on the two target sites.
They are phrased appropriately for the two target sites.
The question posts on the two sites each contain a link to the other.
There are questions that would benefit by being addressed both by a community of experts on Judaism and by a community of experts on something else. For example, a question about the history of some Jewish practice could work here and on History.SE. There are probably people in each community who have expertise in the history of Judaism and who do not happen to frequent both sites. In addition, the history of a particular practice may be different from the point of view of Jewish tradition and from the point of view of the scholarly consensus in the secular discipline of history, so correct answers on the two sites may not be the same.
It is important to make sure that each post is up to the quality and topicality standards of the site that it's posted to. Given that these standards differ from site to site, the author should make sure to be considerate of the particular standards of both sites, especially if one of them is less familiar. Such consideration may result in the questions being stated in materially different ways, or it may not.
Also, note that even the exact same worded question on, for example, MY and another religion site are essentially different questions as they are asking from two different perspectives. Users should consider what kind of answer they seek.1
The question posts should link to each other so that readers of each site who are less familiar with the other site will be able to make easy reference to both communities' treatments of the question. In addition, ready reference to relevant information on another site could inform answers on this one.
1. From a comment by DoubleAA