Yes, the battery fits the Xbox Series S controller.

Yes the battery fits the Xbox one, one s, one x, series s, series x and the Xbox elite 1 controller.

Yes, the controllers for Xbox Series X and S are identical. The box simply contains the controller in the matching colour.

Microsoft Xbox Series X|S Wireless Controller - Robot White

If you still have a Play & Charge Kit from an Xbox One controller, you can also use its battery. You can simply get a matching USB-C cable here.