What's the appropriate notion of equivalence between two objects in a double category?
At first I thought the answer was just an equivalence in one of the associated $2$-categories, but then I realized that objects can be equivalent in the horizontal $2$-category without being equivalent in the vertical $2$-category, or vice-versa. For example you can write down a double category which is a discrete set vertically and terminal horizontally.
So I assume that the correct definition requires equivalences horizontally and vertically, and then some squares relating them. But I can't see the correct requirements for the squares.
I don't think there is a correct notion. A double category is a couple of interacting 2-categories, moreso than a generalization of a single 2-category.
For instance in the double category of categories, functors, and profunctors, which is even a proarrow equipment, a vertical equivalence is an ordinary equivalence of categories, while a horizontal equivalence is a Morita equivalence, in this case an equivalence up to splitting of idempotents. The notions are distinct and separately significant, and it's complicated to see internally when a horizontal equivalence gives rise to a vertical one.