New, based-on Object style isn't "based on"
Take an Object Style and create a new style based on that style. When the new dialog opens with the based-on name present, it has a number of items chosen and selected, but it's not the same as the based on item.
You could try Duplicate Object Style and then add the based on-name, but while there is no error, any change made to the based-on object does not change the new object style.
Based on Style does inherit many of the properties from the Base style. And when you change those properties in the Base style, those properties get propagated to the child style.
Adobe InDesign team
Tim Murray commented
It is close, but not quite. A new based-on (the child) will get the parent's characteristics if one clicks "Reset to Base" when it's created (although one could argue it should not be necessary), but then, at least some options changed in the parent are not propagated to the child. I just changed a "text frame footnote" to "allow overrides" and the setting was not propagated to the child.
Actually, my title is wrong. The based-on part is okay of you click the reset. I should have titled it about propagation. I can delete it and make a new one, if you want.