First of all, that was really funny!
Some dogs have to eat vegetarian dog food because they're allergic to meats. Usually, what happens is this. People start their puppy out on, say, "Chicken and Rice" dog food. The puppy starts picking at his fur and chewing big sores onto himself, turns out he's allergic to his food. So his owners switch him to Lamb and Rice, which is more hypo-allergenic than chicken. Of course, it's not completely hypo-allergenic, so his skin and fur don't recover and he's still itchy and miserable. So they switch him to something a little more weird, like, Rabbit and Potato. The symptoms decrease, but they don't go away. Eventually, once one has exhausted all the possible meat-inclusive dog foods, one has to resort to a vegetarian formula.
Sometimes dogs with Irritable Bowel Syndrome or colitis have to go on vegetarian food formulas too, so they can stop having screaming diarrhea every time they eat.
Although I'm sure there are also some die-hard vegan folk who abjectly refuse to have any meat products in their homes at all and if it's 'good enough for them it's good enough for their dog!' too.

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