Feel free to try both of them out. Sony system has cat/dog eyes detection which is gonna be helpful for pet photography. However, sony sytem is quite expensive considering their better lenses usually cost a bomb.

I currently own a Z6 and if you don't mind the noisy auto focus of the 50mm 1.8G via FTZ adapter, you are good to go. With the FTZ adapter, you still get 3 axis stabilization on f mount lenses. So, it's good for video even when you are just going to handheld your camera for videography.

I am not sure on what aperture you use to shoot on pet, me myself uses F2.8 and above. Even though, Z6 doesn't have eye AF for animals, the touch screen focus is suffice and good enough for me to use. Unless your pet is very hyperactive type then, it's very hard for you to get a nice in-focus shot of its eyes.

In short,

1. Go for Z6 if you do not shoot very wide open aperture on animals.
2. Go with Sony if you can afford expensive lenses and get to use the eye AF on animals.

Hope this helps.