In my field it's often general journal policy, not an individual choice. It's hit or miss, as it can be easy to guess who the reviewer or author is in a niche field. I personally don't go out of my way to figure out the author's affiliation, even if it can be trivial. Regarding self citations, those are usually obfuscated at the review stage. I'd say that a paper is easy to narrow down to a circle of scholars, but it might be the first paper of a research associate, a throwaway paper by a PI, or a paper that aims to engage those narrow specialists. So is a kind of smoke screen.

probably mostly only works for first publication

No it doesn't work.

But it's better than not doing it.

People suspect who the author is but maybe you cited those papers because you're afraid of getting the author to review them, or you're a fan-boying grad student.