I read an article on a group of stars orbiting the supermassive black hole Sagittarius A. Did these stars get caught by the black hole's gravity like planets capture moons, or were they born from the remnants of the star that went supernova and turned into black hole?
They are captured.
The formation of supermassive black holes is rather uncertain. They may have formed from core collapse of very ancient stars (and then grown) or they may have some other mechanism of formation in the early universe. However we do know that the first formed early on in the history of the universe. They are as old as the galaxy, or perhaps even older.
On the other hand the stars that orbit the black hole are of regular age. They are not particularly old. Therefore the black hole formed long long before the stars that orbit it. And so those stars must have been captured by some mechanism.
Star formation doens't normally happen in supernova remnants. A supernova can provide the shock that starts stellar formation in molecular clouds.