Bun,
My first suggestion would be to find something that you can identify with outside off the game. It sounds like you are valuing yourself based on if you win or lose a game in LoL. But League is a game. Unless you are in the top 0.01% in the nation your skill hear means nothing. I mean, you could enter just about any Starbucks in the nation wearing a hat that says "I am rank Bronze 5 in League of Legends" and probably not run into anyone who understands what that means or cares.
All games and sports were originally set up as a form of friendly social competition. It gave people a way to burn off energy and to compete in a controlled environment where the risk of injury was significantly less than in a battle or war. Yes, there would be personal pride on the line, but there was no real harm done if you lost.
Now, I'll admit I can't help much more than to offer my observations. I don't have the same issues you have as for the most part I don't really worry about my rank in LoL or if I'm winning every game. Yes, I enjoy winning, but as far as I'm concerned a loss is rather meaningless. The problem is I can't teach that attitude. It's something I had to cultivate by reminding myself that I had to have fun first.
But I also absorbed that while growing up. My family would have regular game nights where we'd get around a table and play anything from Monotony (my personal nickname for Monopoly) to various card games (usually rummy or a family version of multiple player solitaire) and taking those games too seriously was a way to get banished from the table, something none of my siblings welcomed. So we just didn't take the game too seriously. It was about having fun, not winning every game.
There isn't really more I can say.