We probably all want a pet update,i say it should be called the Paws and Claws update.It adds new mobs,items,blocks,etc.
This is what is i would if i could make this update!
A dog mob:Basically like the cat mob,basically different kinds of dogs.you can still tame a wolf but it will not change...it would be a tamed wolf still.
A bird mob:Similar to a parrot but does not dance and looks different,there would be many kinds of birds,Bluebirds,cardinal,parakeets,cockatiels,robins,etc.
A leash item:it acts like a lead but,it can only work for tamed mobs
reinforced lead:This item is a lead that is harder to snap and break off a mob
Pet beds:this is so if you pets die they will respawn (only cats and dogs) they also sleep in it.
Fish bowl:You can use a fish in a bucket and put it in this glass fish bowl,it looks like a glass block just a little different.
Bird cage:This block you can put your parrots (and/or bird mob) inside of it by get them on your shoulder then right clicking the cage,there is a small bird cage that can fit one parrot (and/or bird mob),a medium bird cages that can hold 2 parrots (and/or bid mobs),and a large cage that you can fit 3 parrots (and/or bird mobs) inside.
I'm not really a fan of changing the pet system to something closer to a Chao Garden from the Sonic series. Pets in Minecraft are meant to fill a purpose which may not always be terribly useful, but could be useful nonetheless.
A cage could be useful on its own for transporting small mobs like Chickens, Rabbits, Foxes, Bats, Cats or Parrots, and possibly Silverfish, Baby Zombies and Slimes if you can manage to catch them. A cage with a bird in it could be a nice decoration too. (I'd say Baby Villagers since that would be useful utility-wise, but that just sounds far too weird and wrong for Minecraft.)
A fish bowl might be nice, but I doubt I'd use it as anything but a fish-bowl. I'm hesitant to add furniture blocks to the game if they can't be reused as something else.
Being able to respawn your pets is great if all you do with your pets is roleplay with them, but I don't think Mojang originally intended for people to do that. I imagine Mojang intended for you to breed wolves as unnamed helpers to help you kill mobs while you're mining or exploring, albeit this is hard to do since they tend to teleport or just walk into danger, or sometimes just right in front of where you're swinging your pickaxe.
That being said, maybe respawning pets would make people want to use them more often. It's hard to not get a little attached to the pets, no matter how utilitarian your playstyle is. Personally I'd do this using nametags, and a pet's respawn point would be placed near the player's spawn point once the player uses a nametag on their pet. Considering nametags have to be fished out of the ocean or found in dungeons, it seems like it'd be a good reward if they made your pets indestructible. (Mind you, you'd want to add a way of un-naming a pet if you need to get rid of them for some reason.)