Dogs is based on their own forms of play, like mock hunting and chasing and those other activities, so I think it's linked to what they see about your fitness in those matters.

Do they engage in something like laughing during that activity?

Both other apes and dogs are social animals with breath control and, at the very least, an "ABC" level of intellect, so within what I'm claiming, it would make sense for them to develop laughter.

It might make sense based on your theory to make a few predictions for different species about which of those species should have something like humor and which don't. If later someone investigates you could validate your thesis.