There needs to be some clarity about terminology here.  Non-human animals do not have rights recognized under any legal system I am familiar with.  This is because they are not recognized to have standing.  Only humans have justiciable rights.  Bambi, even through a human lawyer (and no lawyer-client relationship would be recognized there), cannot bring a wrongful death action against the hunter who shot his mother.

There are, to be sure, many statutes and regulations intended to protect non-human animals from, e.g., cruelty.  But those laws ultimately protect human interests.  Cruelty to animals is offensive to most people; protecting those sensibilities is one purpose of such laws. And you can argue that a human willing to inflict cruelty on animals may be more likely to injure his fellow men.  Litigation has been brought to effectuate such laws, but it must be brought by a human and must assert the human importance of seeing the statute enforced.