It’s actually completely different. The other studios they own are part of Sony Interactive Entertainment’s PlayStation division, but Bungie is on the same organizational level as the PlayStation division, as a division unto themselves.Bungie is 100% owned by Sony.
They can publish their own games if that you saying about “independent” but they pretty much depends on Sony.
It is not different from others Somy’s studios… that how they handle things… each studio has it own politics, hierarchies, CEO, etc.
PlayStation management tells studios like Naughty Dog and Insomniac what they can do, but Bungie makes all their own decisions.