The Shaolin temples taught Shaolin martial arts. Taoist temples taught Tai-chi-Chuan and other internal martial arts. Why are spiritually oriented temples teaching fighting? These martial arts require you to develop your "inner awareness". You must be able to see your behavior patterns that interfere with the sparring. As an example, flinching tenses you up and prevents you from moving smoothly. Unnecessary tension may make you feel strong but it slows you down. And so you must stay relaxed physically and mentally. The ability to shed harmful behavior patterns is necessary both for spiritual and for martial arts training.
These martial arts require that you view the other person not as an opponent but as being part of yourself. You enter into a relationship with the "partner" and try to control the nature of the relationship. In the case of Tai-chi-Chuan ("The Grand Ultimate Martial Art"), you move away from his yang areas (his strikes) and into his yin (open) areas. Thus you don't block his strikes because that would be moving towards his yang area (his fist or foot). As he strikes you immediately move into an open area and deliver your own strike. This is the relationship you establish with him. The result is that you don't feel angry or tense but move with efficiency and ease.
This is an important lesson for spiritual practices as well. Don't view life as an opponent and try to battle it. Work on your relationship with life itself so that your life becomes efficient and easy. In this way the martial arts become just another type of spiritual practice.