Max is a name given to dogs, but it it’s not a dog name. 
I personally don’t understand what people mean when they say that, because most of the names given to dogs are also popular for humans. Just because they are also commonly used for pets doesn’t mean they are exclusively usable for animals.
According to the AKC, the top four most popular names for dogs in 2024 were: Luna, Bella, Daisy, Lucy, Milo, Max, Teddy, and Charlie.
To compare, out of the top ten names for male dogs in 2024, all but three were in the top 200 for boys in the United States, and all but two were in the top 1,000. (The two that weren’t in the top 1,000 were Teddy and Ollie, whose common full names, Oliver and Theodore, were in the top ten for boys).
So, yes, Max is commonly used for dogs. But it’s also commonly used for humans, and I don’t think one negates the other.
I’m not sure if that helps at all, sorry!