Dogs adjust to and 'get stressed from' whatever their owners want them to adjust to or 'get stress from'.
All my life, dogs never minded being 'uprooted', so I don't see any problem. I don't see any dog that gets 'stressed' by going to Mom's during school and off with daughter in the summer. I am very capable, thank you, of noticing stress in dogs. The symptoms are obvious.
If the dog going with her when she can have it and staying with you when she can't is a problem for you personally, admit it, but I don't see any fundamental problem in the nature of dogs that would preclude it moving around. I don't see any problem with the dog going back and forth if SD and SMom both treat it well. Dogs do it all the time. And suffer no consequences. My own dogs did same when I was in college. Quiet time on the couch eating birthday cake with Mom, and exciting times with the daughter going backpacking and hiking.
If on the other hand it makes SMom feel used to just take the dog any time SD wants or that SD is being irresponsible to hand the dog off to SMom now and again, then that's another matter.
Personally, I think the daughter was originally given the choice to either get a dog or not, and was allowed initially to get a dog before college, so I don't see how she is at fault at this point. She was given a choice, she got a dog. Most college housing doesn't allow dogs, and I think kids do better if they live in the dorm or off campus housing than at home. They more suffer the consequences of their actions, and don't continue to rely on their parents as much. Unless the family is very careful with their young adults they get too dependent.
My own attitude toward it is that in a few years, SD will be out of college, and most likely, the dog will stay home with mom, because most cheap housing after college ALSO doesn't allow dogs either. Only either very poor or very expensive housing allows dogs. The trend today is for apartments to be much stricter about animals.
So why get a dog right before college? Most parents wouldn't encourage it because they don't want the dog dropped off with them whenever.
So why does the average young person crave a dog right before college?
Because people get comfort from dogs. A mom alone without the kids might get some comfort from taking the dog for walks and having something to fuss over. Sometimes for a young adult, the only anchor they have is that unconditional love from that dog. If they can be with the dog in summer or on vacation, it gives some shape to their lives and some sort of continuity.
Sometimes a pet is not just a pet. It brings comfort and helps a person make a transition in their life.
And dogs, pretty much, are happy to help out, and plenty flexible enough to live with Mom or Step Daughter.