Not sure about the primary FSW but the BFS uses cFS[0]. As the sibling comment mentions, you can check it out on GitHub. Sadly I believe NASA keeps most of their best code private, probably siloed into mission-specific codebases. Still, the cFS repo is an awesome crash course on old-school Flight Software techniques.
At about 1:20, the presenter says the BFS uses a different OS and hardware (not sure if that means a different instance, or a different class, so to speak).
[0] https://youtu.be/4doI2iQe4Jk?si=ucMoIdw7x_QgZR32