The dopesheet doesn't use the Animations view to decide which skeleton to show. When a project has multiple skeletons, the dopesheet contents behaves slightly differently from normal. Instead of showing all rows when nothing is selected, it shows all rows of the last selected skeleton(s) (assuming the dopesheet is unlocked). It can even show multiple skeletons if you have something selected in both:
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This also shows the cool new skeleton separator we have in the tree in 3.6. 🙂
I suppose we could make the Animations view change the dopesheet contents to the selected skeleton, but only if nothing is selected (else the dopesheet will show the timelines for the skeleton which has something selected (unless the dopesheet is locked of course)).