Companions and wild pets now sport an icon on their nameplates showing what family type they are in, as seen in the screenshots above.
The second pet battle slot now requires the achievement Just a Pup to unlock, and the third pet battle slot now requires the achievement Experienced Pet Battler to unlock. Again, these could just be placeholder achievements. It's a WIP afterall.
Last but not least, the guild vendor has a new pet, Thundering Cloud Serpent Hatchling, but it currently doesn't have a sale price. This could be a placeholder companion since I haven't found its spell or item ID yet, or it could also be a NYI pet. If the mounts are any indication, a Cloud Serpent Hatchling sounds pretty awesome!
Pet battles are still unavailable for most players on the beta, however a few might have the ability to pet battle (if they have the patience to withstand all the possible crashing :P). Last I checked, capturing pets is still turned off as well as collecting experience from pet battles. Those that were able to train for pet battles during the short preview might be able to track wild battle pets still.
It seems like it will be a while longer before Blizzard is ready to allow us to fully test this new feature. I hope it's worth the wait!