Yes, everyone has to apply for a fellowship, even if they are at the same hospital. Some hospitals favor their own residents for fellowship slots, and other hospitals prefer to take residents from other programs, to add to the variety of experiences.
In my residency program, sometimes fellowships would take fellows "outside of the match". This meant that the resident did not have to apply or go through the interview process. The program would not advertise the position, and no outside residents would apply that year.