You are right you can't fight your son's battles anymore. He is growing up and he is entering that transition period between child and adult and there are bumps along the way. At 13 he is entering the world of the teen and emotions and hormones. Some kids are more sensitive than others to situations (my son and grandson) and they will cry at the drop of a hat.
What you have to do is just get to the bottom which was the team losing the game and what to do next time as there will be a next time. As long as they learn from the first to the next to not repeat the exact same thing it will be okay.
Don't go to the coaches. I am quite sure there was a meeting about the game in the locker room which didn't help your son or many others on the team about the outcome. This is part of the world of sports and what people don't see. People cry out of frustration, happiness, anger and disappointment.
The hug and the cookies were "the bomb" mom. Just put this in your book of something you two experienced together for a future chat when he has kids.
Love to you both.
The other S.
PS Been there with the wins and losses of the football world. Have also seen how ugly parents have gotten at coaches and how the coaches have held it all together when they wanted to deck a parent.