Soy sauce is pretty salty. It sounds like a great deal of water diffused out of your chicken and into the marinade, which significantly changed the texture of the meat.
It's not uncommon to do something like this on purpose. When you make gravlax, for example, you cover a piece of salmon with quite a bit of salt and refrigerate it for a day or two. The salt draws out a lot of moisture, causing the fish to firm up considerably. Even though your soy sauce marinade was liquid, it still had a much higher salt concentration (or to think of it another way, a much lower water concentration) than the chicken, and would have had the same effect.
It is fairly atypical to cook large, bone-on pieces of chicken—including breasts—directly in soup. After all, who wants to find bones in their soup? (Making stock is another story....)
Bone on chicken breasts difficult to cook evenly, due to their shape. Some parts are thin, and will cook through rapidly, and other parts are thick and require a much longer time to cook. This makes it easy to have overcooked and undercooked areas. They are also a little unforgiving in soup if overcooked, as they tend to become tough or stringy in texture.
Normally, chicken would be pre-cooked via another method, and then cut up or shredded, and placed in the soup in the last few minutes in order to heat through for service. My favorite method of preparing chicken to use in soup or other recipes is to roast it, still skin on for flavor, but any method will do. The skin can be removed prior to adding the chicken to the soup, if you desire.
Note that this method will not add the flavor and gelatin (for that rich texture) from the bones to your soup. Instead, as cook chicken parts for other dishes, save the bones and other scraps in the freezer. When you have several pounds (a couple kilograms), you can use them to make chicken stock. Homemade chicken broth is a fantastic base for soup, and will help you achieve an excellent flavor.
If you absolutely want to cook chicken breast directly in your soup, I recommend that you de-bone it. Cut it into bite sized pieces. As one of the last finishing steps in preparing the soup, reduce the pot to a simmer rather than a full boil, and drop in the chicken pieces. This will permit the chicken pieces to poach gently in the broth. Your soup is ready for service. once they are cooked through—my guess is about 15 minutes, depending on the size of the pieces, but check them occasionally.
Best Answer
You can't do it, it's physically not possible. Your meat will always taste of meat, not of sauce. Meat is not some kind of sponge which can soak up sauce, it's a dense muscle.
If you want more flavorful meat, you can buy more flavorful meat. Most meat you can get in the supermarket is tasteless, because 1) people don't really like meat flavor if they are not accustomed to it (see the recent mutton question), 2) it's cheaper to raise tasteless meat (young chicken vs. soup hen) and 3) you need a fattier meat to have more flavor, but people prefer lean meats nowadays. Still, the meat will then taste of meat, not of sauce.
If you find your food way too tasteless, you can also consider preparations such as a meatloaf, where you can physically mix spices into the protein.