You're experience is common, but honestly it's not something I personally have. One of my ways that helps me is when I read out loud to myself for 10 minutes or so, and I've always stuttered when I do that. It does help though. It's true I don't stutter around pets, but that's more to do with the funny voice I make, I think. From what I've found out, it's both mental and physical. My theory for the mechanism of stutter is two fold: 1) The neural links between our brain our and speech muscles aren't wired the right way. This inherently causes a stutter, but it can generally be overcome with therapy work/relearning how to talk the right way. The specifics of how our brain is wired wrong are still unknown. 2)Stressful situations cause anxiety which takes away brain power from talking, which means you'll revert to stuttering as a fallback. Hence these situations make the stuttering worse. So it's both physical and mental imo, one feeds into the other.