Hmmm... I think the previous 2 responses simplify this question way too much. Even after we develop abilities to genetically engineer humans, say like from the movie Gattaca, we still will probably be light years away from understanding the mind and indentity.
For a person to understand what and who they are is quite a remarkable feature that isn't really understood all that well. It's one thing for us to engineer an animal with human body parts, it's an entirely different thing for us to 'assign' to this animal an identity, specifically one which we have chosen for it. This is the only plausible way I can see this occurring, if we 'create' an animal which has at the least a human like brain (based on the assumption that most of identity comes from the brain) and we some how give it the identity of a person who wants to live life as said animal.
Before this to occur however we will need to understand cloning and identity problems perfectly.
What does it mean to say I am Kevin? What does it mean to say I am human? I'm talking about the what does the "I" mean period.
The other method which I belief the previous posters have in mind is simply 'transforming' a human into another animal... This will never be possible outside of fictional novels dealing with werewolves and 'animagus'.