I don't think it's a good idea, in the sense that it's a massive floodgate that a judge would be mad to open, but maybe it isn't a bad idea in terms of publicity for the miniscule sanctions handed out to those convicted of serious animal abuse/neglect.
I think the horse is actually named as the claimant in the court papers. So although it can't fill the papers in itself, it is, legally, suing its previous owner.
There have been several other cases and they've all failed on the basis that animals don't have human rights. You'd think they'd work that out before it got into court, wouldn't you