I think this is a very important distinction you make, but it doesn't necessarily mean that they are fundamentally incompatible.
It today's world, data about a person is an asset. A person should own their assets, and have control over them. If there were only one option, this would have to be it - it's the only one that aligns with business interests. If you instead go purely the route of anonymous data collection, because data is such an asset it just gives businesses the strong incentive to find ways to de-anonymize your data. That is an unstable situation, and a societal counter-productive incentive.
Since data on me has value (clearly since businesses are run off it), make it a true product. Give it value, allow me as a consumer to trade it, allow businesses to quantify it's value in the market place, and explicitly bid on. That aligns incentives. As data becomes more (or less) valuable, businesses will adjust their prices for it. This aligns incentives, the more data is worth to a business, the more they'll be willing to pay for it. IMHO this is clearly the right approach.
That said, there is still a world where both methods exist. Users choose when they interact if they want to perform anonymous interaction (or pseudoanonymous), and people provide services to ensure it is anonymous. HN, reddit those are forms people would likely choose pseudoanon for, but there would be a firewall between someones pseudoanon identity and their true identity. And people would be pseudoanon knowing the risks and taking care to not divulge linking identityy info, and businesses would ensure there are limits on data preservation / recording of pseudoanon interactions. The same way people post on HN and use care in what they choose to disclose, HN would also ensur data expiry is short enough and would not share/sell pseudoanon data to 3rd parties.
Again, if you can only pick one, it has to be consumer ownership of their data, but I think they both can work together.