If every person is a person essentially, then the persistence question (what makes a person persist over time) and the identity question (what makes something the same person) pick out the same conditions.
Those who state the persistence question as Question 1 generally do so because they assume that every person is a person essentially: nothing that is in fact a person could possibly exist without being a person. (By contrast, no student is a student essentially: something that is in fact a student can exist without being a student.) This claim, “person essentialism”, implies that whatever is a person at one time must be a person at every time when she exists. It makes Questions 1 and 2 equivalen