In fact, it is difficult to say whether those pet dogs will suffer from depression or mental illness if they leave their owners and are fed by people in pet shops, because each dog has a different personality and situation.
This is the same situation as humans. Dogs and people have similar emotions, but it’s not certain how deep these emotions are, or whether they like to embody them. Maybe some dogs are more familiar with themselves, that is, they are very nervous and don't care. Then there is no problem if you leave them in a pet store for a week or a month. As long as the people in the pet shop don't abuse it, then how it usually lives will probably be the same in the pet shop.
Familiar dogs really don’t need people to worry too much, but it may be different for some dogs. For some puppies who are timid and shy by nature, if they are separated from their owners for a long time, they may develop a special fear, fear, and miss their owners. For this kind of dogs, their mood is not very good, and they may not want to eat, have bad spirits, and are overly sensitive to the outside world.
If this state is maintained for a long time, then when the owner goes to the pet store to pick up his puppy, he may be too thin and then fall ill. It is also possible that the puppy will no longer recognize the owner because of no contact for a long time. It is not impossible. Because for animals, especially dogs, once they develop a sense of dependence, it is a sign that they cannot live without you.
So if you don't communicate with the dog in advance, or if the dog doesn't know that you will leave it without an owner for such a long time, and directly gives it to others to feed, then for some more loyal dogs, there may be some hunger strikes, or depression, or in more serious cases, death.