Bacteria aren’t really plants or animals. They kind of fall into their own seperate little area. They have characteristics of both really. Some photosynthesize, while others get energy from from other chemical pathways, similar to digestion. I say call 'em what ever the heck you like.
Back to the OP though, I am one of those who are just a bit lactose intolerant. I don’t get horribly sick, but the smells that I make could gag a maggot. If I avoid most raw dairy, I’m usually good. Milk is the worst. Cheese/butter don’t usually bug me. Cooked dairy doesn’t bug me at all.Get a glass or two of milk in me and watch out.