I recently had the privilege of working with a dog/handler team that had particularly good interpersonal play skills. Without using food or toys, this handler could engage her dog for several minutes at a time. Further, she could then ask the dog to work for her in this engaged state, which allowed her to space out her food rewards significantly. I was impressed because I consider "interpersonal play" to be a strong indicator of trust and mutual enjoyment between dog and handler - simply you and a dog interacting with no intermediary. I work hard to get to this point with my dogs, and it's one the most difficult things I do in training.
Imagine my surprise when this same team was criticized soon thereafter for showing "no relationship" in a training situation. The trainer's explanation was that the handler could not play tug with her dog for extended periods of time, and this indicated a "relationship problem".
Playing tug is a mechanical skill. I frequently play tug with dogs in seminars - more often than not I don't even know the dog's name. Does this mean I have a relationship with the dog, simply because I can keep a dog engaged with me when I hold a toy? Of course not. It means I have really good mechanical skills with a tug toy. I can do the same thing with a cookie - as long as the dog KNOWS that I have the cookie, I can keep many dogs engaged with me. That is no more a relationship than if I tie a tug toy in a tree and teach my dog to grab hold and tug- surely we would not say that my dog has a relationship with the tree?
Games that involve toys DO help dog/handler teams build relationship, because it is almost impossible to play tug with a dog and not get involved personally. Good tug is a fine game of give and take; the handler studies the dog very carefully to figure out exactly how best to play with that dog, and the dog learns the rules which cause the handler to interact. Over time, the dog develops a great appreciation for the person who is able to play so well and who engages the dog in a game which is a whole of fun. But it's not the tug itself that is building the relationship; it's the interaction between the dog and handler. The toy is an intermediary, a useful but tangential tool.
A person can use food in exactly the same way to build a good relationship, but it's more difficult because lazy feeding is common and easy; no energy or genuine interaction from the handler is required. Using food in training tends to create a strong relationship between the dog and cookies, but the owner may or may not be seen as significant. On the other hand, if you have the dog chase you around and jump for each cookie while you cheer and interact, then food play has the same effect as toy play; it builds the underlying relationship.
I was saddened to hear that this handler was distressed after that seminar; she no longer felt good about her relationship with her dog. What I had seen was special and unique. I felt good when I watched them; I looked forward to the day when my young puppy would see me as a wonderful person - whether or not I held a toy or a cookie. While I agree that it is in the handler's best interest to work on developing her game of tug, the purpose of the game would be the energy it produces, not the underlying relationship. She's already got that.
A great relationship is based on choice; the dog chooses to spend time with you because you are important to him. He feels good when he is near you. He wants to hear your voice. He is happy when you pet him gently; not to play a rousing game, but because he feels valued in your presence. Relationship is expressed when you let the dog out of a crate and he greets you before the other dogs - not because he is trained to reorient when let out of the crate, but because he genuinely enjoys your attention. Relationship is a dog that tries to get you to interact; the dog who brings you a stuffed toy not necessarily to engage in a game of tug, but as a gift. Relationship is the dog who looks to you for guidance when he is unsure of what to do or how to react. Relationship is not what you provide for the dog; it's that special feeling and bond that develops over time, as the sum total of everything you are together. The moments you spend training, playing, and living together - these build your relationship. If your dog interacts with you only because you provide food, toys or freedom, then I'd argue that you do not have a relationship; you have a dependency.
Simply put, relationship is not food and toys; relationship is what's left when the food and toys are gone.