Not always—it depends on the publisher for sure, and possibly the field (e.g., physics, chemistry).

In biology, you have several models for peer review. Completely blind reviews where both reviewers and authors are anonymized. You also have semi blind models where the reviewers know the identities of the authors, but the authors don't know reviewers' identities. You also have open reviews where everyone knows one another's identities.

In completely blind and semi-blind models, you occasionally have reviewers that reveal their identity.

She was told to read and cite the other work. I take that as meaning she hadn't intended to use her previous work as a source, but they wanted her to

It's a catch-22 situation. You are supposed to disclose if you wrote the thing you're citing, but also cite in third person, and also it should be obfuscated for the peer review. So, what happens is that you write something like "in the author's previous work (yourownname, 2017)…" then that gets censored by yourself or whoever is in charge of the peer review, "in (blank) previous work (blank)…". Now, if you're experienced in reviews you can probably guess it is the author of the paper you're reviewing quoting themselves. But you still don't know who it is, and you could never guess right whether it is Ruth Gotian or not. So you're back to the tweet's situation.