In the novel Persuasion the identity of the family is defined by their peerage, name, and precedence in rank since they literally belong to the upper class. This being said, Sir Walter has given the family a collective personality based entirely on the history of the Elliot clan, and their territory.
As a result, the individual members of the family have become a by-product of the family name. This is not a bad thing necessarily, but just a point that they have become Elliots before becoming individuals. The servants and family members have been brainwashed by Sir Walter to hail the Elliot clan, to live in a way that befits the Elliot clan, and to pretend that the Elliots are perfect.
The effects of pressure of this kind is that chances are the individuals might get diluted within the idea that Sir Walter is trying to instill in them that the Elliots are untouchable, and their personalities might end up depending entirely on what happens to the Elliots, as a whole, and not to each of them as individuals.