Why is peer review anonymous?
The big reason for anonymous peer review is to allow the reviewers to speak their mind without fear of the authors taking revenge. Even if it's easy to identify, retaliation can happen: it's already decently common for authors to get angry when their paper is rejected for reasons they do not agree with (example). Even if the authors don't retaliate this obviously, future relations with the reviewer can easily be strained: for example, say the author and reviewer communicate several times and eventually the author decides that the reviewer's an idiot and submits elsewhere. If the author later finds themselves reviewing the reviewer's grant proposal, it will be hard for the author to remain objective.

Another issue with non-anonymous peer review is, the signature of a senior person as a reviewer can carry the implied threat of "do what I say or I will crush you". The consequences can be unpleasant for the authors, especially if they're junior. Example.

Yet another issue with non-anonymous peer review is that it's not clearly advantageous. There is no strong evidence that open peer review improves review quality (or makes it worse for that matter). Instead there was an immediate negative effect in that more people decline to review. If it's the case that there's no advantage, then implementing a fix seems rather unnecessary.