You should be able to have two or more pets out at once.

I play solo so it doesn't matter to me

just let me mute yalls pets lol

I fought some brig a season or so ago with 5 pvpers on the boat and they harpooned me on their boat and I was jumping around while they were shooting at me. They all were in the same outfits and they all had poodles out.

Eventually they got me and all of their poodles went wild with the barking.

I'm gonna need that mute lol.

@watertowerkid
If I counted correctly, there are 150 different pet variations (not including outfits)
That means one galleon could have up to 600 pets on it at once. How would that run? And furthermore, how would that sound?
Now, obviously this is a worst-case scenario, but the point is that completely uncapping the pet maximum is a bad idea for performance.
Increasing the maximum to 3 seems reasonable, but a maximum of 3 still means servers could have nearly 50 pets at once. In a large scale PvP battle, performance would suffer terribly (The game sometimes struggles with 50 chests, and those don't even move) and it's a much more realistic number for a server to actually reach. Plus lag would spike whenever large amounts of pets teleport onto islands or into the server via a merge.
Not to mention galleons with 12 pets will have trouble interacting with things between all the cats and parrots.
Honestly, increasing the cap to 2 would be okay with me. That being said, I don't have a lot of data on how large numbers of pets affect server performance.
On an unrelated note, I quite like @WolfManbush 's idea to mute pets!