I don't think so. In my larger version of the photo, you can see that the book Olga is reading has a border all around the edges of the cover, and does not have a marbled finish. The rectangular mark on the back cover is also repeated on the front -- it's part of the cover itself, not a label. (The children each had unique bookplates with their initials on them that were pasted inside the front cover of their books, so there was no need to put labels on the cover.) It really doesn't match the eBay book at all, though Olga did have a copy of The Princess and the Goblin, which her aunt Irene gave her in 1903.
BTW, many of the imperial family's books were specially rebound in leather, so it may not be possible to ID them by comparing them to the original editions.