Quick back of the envelope calculation to drive home your point:
There are ~300 million people in the US. So you can be identified with 28.2 bits of information.
There are ~42000 US zip codes, so knowing that counts for ~15.4 of those bits. Birthday (ignoring year) is worth ~8.5 bits, gender is worth one bit. (plus a little if the database includes more than two options) So that’s 24.9 bits of information right there; only 3.3 bits remain. So, under the naïve assumption of uniformity, we would expect there to be only 9 or 10 people fitting those constraints. Almost any other fact about ones self would allow complete identification.
