What is the intensity level of american kickboxing?

This is just me, but Kickboxing would be much more intense as your legs are being used and this causes the heart to work harder because the legs have bigger muscles.

Well Blake that's really a toss-up. Kicking does take more energy and stamina so that could be a factor in this along with the level and intensity of the class and instructor. That last part is really the biggie in all this I think and you should maybe go and watch a class and talk to the instructor afterwords about your problem. Perhaps you can stay in the back row or line on drills and sit out some when needed as any instructor will not want you fainting or passing out or having an asthma attack right there during class.

Also sometimes other things can become factors like was the boxing gym air-conditioned or not and what did the coach and instructor have you doing as a beginner. I can make my school pretty uncomfortable by turning that air conditioner off and threaten to when students are not working hard enough in general. Would I actually do that though; probably not especially if someone like yourself is in the group. I can also work students to the point where they will get sick but I don't. Did the boxing coach not take a gradual approach to you starting and build you into better shape and work you up gradually to doing things or were you just expected to keep up with everyone no matter what?

Look for something where the coach or instructor employs some common sense and will maybe make some allowances for your condition when and as needed and that you can handle. That is key to the answer to your question I think more than anything else.

Different trainers, train differently and make it harder or not as hard etc. But if your gonna join Kickboxing or Boxing you should just tell the trainer that you can't do anything to high level because you have asthma. Also they always say that you should consult your GP or Doctor before starting any exercise so maybe the doctor could tell you what would be good or bad for you

It varies by instructor. You'll have to check out the schools around you and talk to the instructors about your asthma. They'll let you know of they think it will work out for you.