Uh...not so much, sorry Sig....In order to legitimately consider oneself a Buddhist, one has to belong to a particular Buddhist tradition/school...
Buddhism can be considered a life philosophy OR a religion depending on the exact nature of the Buddhism one practices. Like any other school of human thought, Buddhism ranges from conservative semi - Hindu religious practice replete with mantras, reincarnation, the wheel of the Mandala and hierarchies of spiritual purity on the one hand to a practical day-to-day way of life totally devoid of spirits, karma, gods and the like on the other.
I am a Buddhist, my practice is NOT religious. I have studied koans for decades as well. They have been distilled for centuries with the intention of teaching through contemplation for the betterment of the student. While they can be helpful, they are not really even necessary, as life gives us koans to consider on an ongoing basis. My presence here and my contribution to this thread are a koan of sorts. If you consider what I have said and profit from it in some manner, then the goal of a koan has been achieved and you will have learned.
Yes, the (relatively secular) Jesus Schools taught with parables that are very similar to koans. When the Jesus Schools had to join up with the Christ Cult in order to survive, the blend of the two sort - of messed that aspect of the teachings up. The Qumran scrolls retain that koan - ish flavour though.
Example of a traditional koan:
2 monks were on a journey when they came to a river. The bridge had been washed away, so the only way to cross was to go through the water.
A woman was standing at the bank looking across with consternation, wondering how she was going to cross. The elder monk offered to carry her to the other side on his shoulders. She accepted his offer and he carried her across, setting her down on the far side safe and dry.
The monks continued along their way, but the younger monk was very upset and let the senior monk know this.
"How could you not only touch a woman, but carry her across the river when you know full well that we monks are not allowed to do such things?" he asked.
"I put her down when I reached the far side of the river, you are still carrying her" responded the senior monk.
Another example - one of my favourites:
A monk was trying to cross a river, walking along the bank looking for a bridge but finding none. He saw another monk across the river and called to him "How can I get to the other side?".
After a moments thought the other monk called back "You are on the other side!"