"Speaking likenesses" is a little-known book by Christina Rossetti, published in the late 19th century. It was never reprinted and I haven't managed to get a copy, but it's supposed to be inspired by "Alice's Adventures in Wonderland", but rather didactic. The book contains stories of girls (some good, some bad) who get their just deserts. I don't know much more than you can read here
The only meaning of "speaking likeness" I could find in the Internet is something like "way of depicting people that displays them in the act of speaking, starting to speak or having just spoken".
Does the phrase have any strict meaning? If it doesn't, what meanings could it evoke in a late 19th century reader? I don't even know if it's more about "a likenesses that are speaking" or "speaking that is done by likenesses".