Play Fighting? Or fighting to play..

As we can see the toy above my brown dog’s head, Im wondering if they’re play fighting, or fighting over the toy.

They do this behavior without any toys, having “zoomies” throughout the house and then doing this to eachother.

If Im correct, the exhales of air signify play?

My brown dog is kind of a “bully” hence my concern of if they’re actually playing.

Additional context:
Cavapoo(white)- 1 year old intact (had the toy first lol)

Chihuahua Mix (brown) - 2 years old intact

Chihuahua was 1 when cavapoo was brought home (he showed bad symptoms of a broken bond due to his sister being rehomed before we got him, hence the brother lol)

Otherwise they walk together fine, sleep near or together, etc…