One time I was browsing through the toy section at a pet store and saw a tug that actually had a warning which pointed towards this myth.
The reason I believe this myth is still working its way into the minds of pet owners is because most pet owners lack the concept of a proper relationship between a dog and its human; which can create a lot of issues like dominance by default or pushy behaviors. And if a game like tug of war is part of the routine, people are likely to want to point the finger at it. In this case, it’s a game in which the dog gets to show its power, and in some cases, even vocalize during the game (which is not necessarily a sign of aggression or dominance). Some dogs just get so into the game that they growl while playing. To the pet person, this growl may look like a sign of aggression or dominance, not understanding what’s really happening.
Playing tug can actually be a great stress relief for your dog. It’s also a great tool to build confidence and establish boundaries. The game should be fair; you should let the dog pull and then you pull back. It should be a game of give and take but ultimately you should dictate how the game ends. If your dog does develop a dominance problem or aggression, it may be something else that is creating the problem, not necessarily the toy or the game.
It’d be similar to playing video games with my kids. Some might say doing so will decrease the level of respect in which they see me because we’re at the same level when we play those games. This however should not affect the way my children see me or treat me. If my children lose respect for me, it wouldn’t be because we played video games together, but rather because I have failed in other areas of parenting.
If your dog does have an aggression problem then by all means don’t give the dog an opportunity to display it, especially if the aggression seems to appear when a toy is involved.