As you say, the main use of category theory is to organize very different areas of math in one overarching framework and generalize ideas from one area of math

As you say, the main use of category theory is to organize very different areas of math in one overarching framework and generalize ideas from one area of math to others. It might be used in the pursuit of developing AI but it is definitely not "an approach" to developing AI, just like "taking ideas from books" isn't one.