This one has a simple answer: no, eggs are not a dairy product.
What is dairy?
Dairy products are defined as anything that comes from a mammal's milk. That includes milk, butter, cheese, yogurt, and ice cream. The milk can come from a cow, a goat, a buffalo, or even a yak.
Why does it matter?
The question of whether or not eggs are dairy is important because many people with food allergies or intolerances often have to avoid categories like dairy, so it's critical to have a clear understanding of what's in the category and what isn't. Some people suffer from an allergy to the proteins found in milk. Others are lactose intolerant, which results from an enzyme deficiency that makes it impossible to break down the lactose (a sugar found in dairy) in one's system.
People with one or both of these conditions don't necessarily need to avoid eggs as they are devoid of milk proteins and lactose. Eggs are also gluten-free.
So why do eggs live in the dairy section?
There are a number of reasons for this, dating all the way back to the early days of the corner grocer. Both are animal proteins, so they make a logical pair in that respect. Both are required to be refrigerated in the United States, so there's a practical reason to co-locate them in the cooler. And finally, many years ago, when local grocery stores were supplied with eggs and dairy, they often came from the same small farms––so it just made sense to put the products next to each other when they arrived. And once the pattern was set, it stuck.