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Creative Commons calls these "free" vs. "non-free" cultural licenses, explained on this page. On dealing with data, CC says: "The data or other contents contained in the database are subject to copyright if they are sufficiently creative." You'd just have to prove that any data you want to put a license on or copyright is "creative enough" which seems fairly subjective and gives you some legal wiggle room.
Creative Commons also mentions (on this page, specifically for data):
"Where CC0 is not desired for whatever reason (business requirements, community wishes, institutional policy…), CC licenses can and should be used for data and databases — with the important caveat that CC 3.0 license conditions do not apply to uses of data and databases that do not implicate copyright."
It might be worth noting that right now, the big data repositories offer CC licenses of varying degrees (Dryad, Open Science Framework, Zenodo, and more). You have the right to embargo data or set permissions for use if there are privacy concerns. You can deposit your data with a repository and ask that people who want to use it have to email you and explain why/for what purpose, and then you can send them the data or deny the request. The Qualitative Data Repository has excellent examples of this use case.