My interpretation of what you wrote is that because the outlets are three prong, you "know" they are grounded. That is not the case by any stretch.
There are thousands of 3-prong outlets that are installed with NO wiring connected to the ground at all, and there are thousands that have a jumper between the neutral and the ground on the outlet. Neither of those approaches are safe or up to any code.
And since you mention that the two outlets are two different styles, one 15 amp and one 20 amp, and you noted that they both were tree prong, I'm thinking you may have some two prong outlets in this house as well. You've found a 6-gauge wire connected to the cold-water line to your washing machine, which is not usually where the ground for the main electrical panel is found. It is normally connected to a primary water line near the electrical panel. And since you did not mention that your electrical panel is close by and are questioning the wire's purpose, I'm thinking the line to the washer is not particularly close to the panel.
So, while that wire could end at the electrical panel, it just as easily could end at a row of three prong outlets. #6 is pretty big wire, but I've seen where good meaning people have installed a large wire as a ground path for new outlets where existing two wire power and neutral cables were reused. Grounding them to a cold-water line isn't per code, but it does make the outlet checker read correctly, and is safer than using a jumper between the neutral and the ground on the outlet.