What a great question! Too bad it has a complicated answer!
In the way that the word is defined in English I would have to say "no," My Life in Dog Years is not a biography. Most usages of the word restrict the use of the label "biography" to mean the story of a human's life, not a dog's life. It would be closer to an autobiography for the sections in the descriptions of the dogs where the author includes stories of how the dogs influenced his life.
The tricky part would be if you went back to the origin of the word "biography." Bios=life, graphia=writing. In the Greek sense of the word, My Life in Dog Years could be considered a biography because it does not implicitly say human life.
I suppose it is up to you, then!