Apparently the parrots got along fine using FB Messenger to call each other, so rather than a distinct "animal internet" it's more "animals on the internet". I suppose that's what happens when you invest effort in designing UI that a toddler can use.
This is why years ago I bought the domain www.omeowgle.com that I never got something working for at the time. I was going to setup a video call system for my family to call my cat (and maybe eventually some friends cats too). Now I'm just hosting some social media for my cat on the fediverse there. https://www.omeowgle.com/@buzz
I have totally wanted to build out "pet VR". Put an animal on a type of treadmill in a virtual world and let them run around. The major issue with the idea is that dogs rely on smell quite a bit so it would likely not be a decent replacement experience for them versus going outdoors.