No there isn't dairy in pizza by default since a pizza is a oven baked flat bread, and perhaps you didn't know but there are many vegan alternatives to cheese and dairy in general.
I don't have kids, and might not conceive any in the upcoming years. Who knows how much society in general will advance in the coming years. Besides that I wouldn't let my kid eat over at a stranger's place, so I'd make sure they knew exactly what my kid would eat and what not. Also again there are many vegan ice creams and candies, and chocolate by itself is vegan, as long as you don't include milk. It's a misconception that vegans are somehow limited in what they can eat, it's just that society has taken the use of animals so for granted that their products are incorporated into many many things, often unnecessarily.
As for the drug analogy, it's to highlight the irresponsible habits of many kids and the parents that condone them. There are enough (older) kids that do drugs and alcohol, so nothing weird about that. It's not only about prevention but also about educating your kid on moral and health issues. Besides that animal products aren't only a tradition for many, but perhaps more so an addiction.