Play between dogs on a dog walk is usually primal
When you think about the evolution of dogs, that 20,000 years ago they were wolves hunting in packs in the forest and now the majority of the ones we see in city areas are just human play things - called companion animals or domestic dogs - they have come a long way.
That evolution means that there is a lot of complex things going on in their head, consciously or subconsciously. For instance only a few breeds are rarely interested in hunting, so even the most meek of dogs, on a pack walk, is probably looking out for something to catch and eat.
Dogs are also a very strong pack animal but also usually need to know where they fit in that pack. If you walk the same dogs in the same pack regularly and they see the same dogs at the park, they won't usually try and change the status of that pack order or where they live in the pack.
But did you know that almost every play that a dog has is about trying to increase its order in that pack. It doesn't have to end badly as social dogs love to play too, but very rarely is there a play that doesn't have a substantial amount of serious rank sorting involved in it.
The miracle with social dogs, and that we chose such a great companion, is that dogs can quickly sort out the order in the dog park, and just continue on walking like nothing happened. No bruised egos no continual attacks on each other. Some dogs will want to play with every other dog in the pack even though they are an adult. Sometimes this is a sign of immaturity on behalf of that dog, a not fully formed personality or a dog that still thinks its a puppy and has to keep refining its skills. As long as this is done friendly and they eventually give up, or don't push it too far, then the pack wont really mind, it just adds a little spice to the day, keeps things interesting.
If only humans could learn that things can get sorted so quickly and for the benefit of the pack, keeping that order can be vital. Imagine humans being regularly so selfless for a team without money being involved, just imagine it/