I've worked in a large company that has "lots of medical data" which we used for analytics purposes. What we would do is "Anonymize" the data. Unfortunately that's a pretty herculean task.
Now, I don't understand DP well enough and information theory/signal processing still seems a bit like "dragons be here" to me. But, I want to take a stab at trying to reason why he said that.
For example, take randomized response (the only DP technique I understand). That is vulnerable to a longitudinal attack: a person can query repeatedly to wash out the randomness. If you think about it, isn't it the almost the inverse of a repetition code (error correction)? There, you're trying to use redundancy (repetition) to remove noise.
If your signal processing professor was already taking that into account then I would be curious to know how that attack would work.