As the other responders have mentioned, it depends on the severity of the allergy and the age of the children. In daycare and pre-school, where the kids are too young to be responsible for their own allergy safety, it was common for my kids to be in rooms where there were no peanuts and/or tree nuts allowed at all. As the children get older, the restrictions tend to get less strict. For example, there is a "peanut free" table in the lunchroom because that is so common and any children with rarer or more severe allergies is accommodated on an as-needed basis.
Last year (1st grade) the classroom was also peanut free, even for individual snacks, which is fine IMO. There are plenty of healthy peanut-free snacks. This year (2nd grade) the classroom is only peanut- and tree nut- free for shared snacks and special events, but we've been living peanut-free for snack time for so many years that I wouldn't send in a peanut or tree nut snack anyway. No need to stress out the allergic student or teacher.
Dairy is a new one to me - most people I know who are allergic to dairy are only allergic if they ingest the items, but if you have a child who is allergic to touch or smell of dairy, then yes this may mean no dairy in the classroom. It's not the end of the world - presumably, even if there is no dairy in the classroom lunch will be less restrictive and you can certainly find a wide variety of non-dairy/nut-free snacks for snack time. Fresh fruit or a baggie of baby carrots and a bottle of water make excellent morning snacks.
And finally, it's been my experience that schools don't take food allergies lightly and only restrict a classroom if it's needed. So yes, this means that to preserve the health (and possibly, life) of one child the rest make accommodations. It's not a ridiculous idea and one that we all need to get used to as food allergies are getting more and more common and need to be dealt with among adults as well. If your child doesn't have an allergy, consider yourself lucky. I'm sure that the parents of the allergic child, and the child him or herself, would love to trade places with you and your child. They're not doing this to get attention or be pains, they are requesting accommodations because the child needs them. Have some sympathy and compassion and make their lives easier. Hopefully once you get the information from school, you'll have a clearer picture of what the restrictions actually mean to your child's classroom.