Great question! In my opinion it depends on the situation and the animal as well. Some animals are taken out of terrible situations and they are so relieved to be out and in a safe place, even if that place is a shelter. They are not in the painful situation they were in, it’s not always ideal but it’s better than where they came from.
Some animals I’ve spoken with had a strong knowing that they were not with their forever people, that the people who were taking care of them were a “rest stop” to where they really felt like they were being called to go.
Some animals who are rehired from loving, happy homes to some one else’s (for a variety of reasons not all are bad ones) can have emotional issues because they are feeling confused, or worried that they did something wrong to make their people give them up, but usually they stop feeling that way, when they realize their new home is REALLY their forever home.
Some animals have a very hard time adjusting and need outside help to adjust either from a communicator or behavioralist.
Animals, just like people are not all the same, so a “one size fits all” attitude is a very big mistake, many issues can be missed when coming from this viewpoint.