In short: it's not anonymous.

They're using a hash function on personally identifying information such as names, addresses, DoB and phone numbers, but Facebook and LinkedIn have enough data that they could work out what hashes correlate with which names, addresses etc. , which would enable them to correlate the hashed data with a specific person that has that data already, and from there they can correlate the hash of the data they don't have for that person with other people in the data that they do have the data for to add more data for that person.

e.g. Someone left NZ in 2015, but hasn't logged into Facebook since 2010, so Facebook doesn't have any up to date data on them, but if they run thier name and DoB through the same hashing function that the IRD used, and say they find one result, then they can update thier database with the persons new data from the IRD.

They just need to find users in thier data where there's only one result for each of the resulting hashes, and can also create new entries in thier database for people who've never even used Facebook but were in the data the IRD provided.

To understand the specifics you'd probably need to do an OIA request or something IDK.