Wrestling is a self defence martial art and also an (Olympic) game which audiences sit and watch. Other martial arts like karate are the same - with a martial/self-defence and sport component, so why is e.g. karate only considered a martial art and not also an Olympic sport?
TL; DR: It has to do with the fact that they have separate international governing bodies (see breakdown below).
The Olympics have a tiered classification system, that can be a bit confusing if you look at each different thing as a "sport", like you would for general consideration. These tiers are:
- Sport - Top tier, and there are limitations on the number of sports that can be in an Olympics. A sport is governed by a single international federation.
- Discipline - A classification underneath a sport.
- Event - Competition that leads to awarding of medals.
So, for example, take the sport of Cycling. It is structured as follows:
- Top level sport, governed by the UCI (Union Cycliste Internationale, or International Cycling Union), which governs all cycling worldwide. The sport of cycling includes 4 disciplines:
- Track cycling (Men and Women)
- Mountain Biking (Men/Women)
- Road cycling (Men/Women)
- BMX (Men/Women)
Individual races in these disciplines are events:
- Road race
- Road Time Trial
- 3000m pursuit etc etc.
Each of these events will lead to a medal ceremony.
The key points of the above, is that every aspect of the above is governed by the UCI. There are individual governing bodies within the nations (USA Cycling for the USA in this example), but they all are under the umbrella of the UCI.
If martial arts wanted to be listed as a sport, and encompass wrestling, boxing, karate, tae kwon do, etc., then the governing bodies for these would need to either dissolve and reform or otherwise belong to a single international governing body. This in fact is how snowboarding was included in the Olympics, the Olympic Committee mandated that their international governing body dissolve and become part of the FIS, so that adding snowboarding would not increase the number of sports, just the number of disciplines.
Note: As of Tokyo 2020, karate is an Olympic sport.
Whether a sport is recognised as an Olympic sport is ultimately up to the International Olympic Committee (IOC). To qualify, a sport must necessarily:
- conform to the terms of the Olympic Charter
- have an international governing body recognised by the IOC
- file an application for inclusion through the IOC
The application process for inclusion in future Olympic games involves "a recommendation by the IOC Olympic Programme Commission, followed by a decision of the IOC Executive Board and a vote of the IOC Session".
Note: JohnP explains the IOC's distinction between "sports" (e.g. wrestling) and "disciplines" (e.g. freestyle wrestling, greco-roman wrestling) in his answer.
Current Olympic combat sports
Additionally there are the non-combat but still martial-derived target sports:
Demonstration summer sports
Non-Olympic combat sports with governing bodies recognised by the IOC
The following combat sports (if eligible under the terms of the Olympic Charter) may apply for future inclusion in the Olympic games (as happened with Karate for Tokyo 2020):
Historic Olympic combat-related sports
The following sports were historically included in Olympic games, but are not currently recognised:
- hoplitodromos "armoured foot race"
- pankration (- 1900)
- singlestick (- 1904)
Wrestling is a game because it is a competitive sport.
As to karate, there are two kinds:
One kind of karate is a martial art - and such styles and schools which teach this kind do not focus on the competition - they focus on self-defense (occasionally, they may use sport as a means to that end).
Another kind of karate is purely a sport - a game. Just like wrestling.
And like karate, the same is true with Kung Fu. Some is sport, some is not.
And as has been pointed out, karate enters its Olympic debut in Tokyo in 2020. So, that kind of karate is probably sport. It remains to be seen what that will look like, but if it looks like a lot of flashy stuff, doesn't incorporate weapons, and its sparring looks more like a match than a self-defense exhibition, then it won't be applicable to self-defense, making it a game.