Typically the publisher will pinpoint one researcher because of her/his expertise, and ask her/him to perform that duty. I have done so on several occasions, and I doubt that you can simply volunteer. Publishers are keen on having reviews done quickly, whence they will NOT wait until a volunteer come along.
As much as I have seen, book reviews are done in each journal by different people. In the bibliographic databases I read sometimes several reviews of the same book, written by different people and published in different journals. I did not see several reviews published in different journals by the same author.