"People ask me what I do when I want to relax and are often baffled when I tell them I race motorcycles. They don't seem to understand that when you're traveling 120mph around a corner with a knee a quarter inch off the asphalt, if you're not relaxed, you're @!^&ed."
The things which relax people are often baffling. I, for one, do not understand how running long distances can ever be considered relaxing, but people do it!
I would argue that there are two reasons why a game like chess is not relaxing. The first is the competitive nature. If you focus on winning the game above all else, you will naturally enter a stressing mindset. For some environments, such as tournament play, this is desired. In more casual games, it can be undesirable. It can help to reframe the game not as a way to compete with the other person, but as an opportunity to better yourself, sharpening your claws within the rules of the game.
The other reason is that you use stressing approaches. This can happen to anyone, and can go completely unnoticed. We often have many different ways of resolving a situation, each of which taxes the brain in different ways. For example, we may be able to use a gut instinct approach to play where you slowly look at the position until it makes itself clear to you, or we may use a highly analytical approach which involves keeping mental maps of game trees and board positions in our head. Choosing the latter will add to stress, no matter how much you focus on bettering yourself.
The Zen solution to this is not just to focus on bettering yourself, but to focus on letting the game make you more aware of what is going on inside. When you realize that you are using a particularly punishing approach to a position, revel in it. Use it as an opportunity to learn how to use that tool. In general, this leads to a decrease in stress as you home that tool into something which is just as effective, but far less stressful.
We do see a pattern in human nature for relaxation. Many people who learn to relax while doing an otherwise stressful action teach themselves to strive towards something which can only be attained by relaxing. Through this, they gain the skill to relax anywhere, and that is a powerful skill.
There are people who are at their calmest jumping out of perfectly good airplanes. I'd call them freaks, but who am I to judge. The things which relax us are tremendously personal. I recommend exploring them.