To my knowledge, full book manuscripts are never reviewed anonymously. Given that the double anonymity of peer review is implemented to decrease biases, and presumably, thereby increase the focus on the quality of the writing, this is puzzling. David Chalmers wrote, in a very helpful comment on how to publish a book "Most book refereeing is not blind, unlike journal refereeing. And when what's being reviewed is a proposal rather than a full manuscript, reputation of the author make a huge difference in reviewers' and editors' confidence that the proposal will be fleshed out well to a book."
While I can see that the reputation or renown of an author can relevantly play a role at the proposal stage in assessing the competence of the prospective author in writing a full manuscript, I don't see why it should play a role when the full manuscript is reviewed. This will inevitably happen when review of full manuscripts is non-anonymous. It would be hard not to be influenced if the author of one's manuscript happened to work at high-ranking institution, is very senior, and already has an excellent track record (I declined to review a book for a major press for this reason), or conversely, if the author is relatively junior, working at a teaching-focused or obscure place.
One could argue that the prestige of an author, or lack thereof, is more important in evaluating book manuscripts compared to article manuscripts because it affects the potential marketability of the book. However, the commissioning editor can take that already into account. It is the reviewers' job to focus on whether the scholarship in the manuscript is sound. They need not worry about marketing.
So I would suggest that while it's reasonable to keep the review of book proposals non-anonymized, full manuscript reviews at academic publishers should ideally be anonymized. Having reviewed several books, I can say it would easy for publishers to make this happen (I would not have been able to guess the authors' identity) and it would increase quality control, as well as taking away an extra hurdle junior scholars face to get their first book out.