You're right that with repeated sampling you can learn more about the data set. If I understand it correctly, I think the solution to that in DP is that you have a limit to how many times you can make your repeated query before you have spent your privacy budget and are cut off. The idea is that for a given budget level you are limited to fewer queries than you would need to learn enough to de-anonymize the data set.

If your signal processing professor was already taking that into account then I would be curious to know how that attack would work.