One popular trope is the 'fish out of water' story. For example, Percy being in the Kung Fu Panda world not understanding all the martial arts and cultural norms there. Or Po in the world of demigods, confused about the Greek mythology stuff.
The 'culture clash' trope is also prevalent. Since the two universes are so different in terms of culture, mythology, and the way things work. In fanfics, we often see characters from one universe being shocked or intrigued by the elements of the other. For instance, the demigods might be amazed by the talking animals in the Kung Fu Panda world, and the Kung Fu Panda characters might be bewildered by the magic in Percy Jackson's world.
Another common trope is the 'team - up against a common enemy'. Writers love to create a big bad that threatens both the Percy Jackson universe and the Kung Fu Panda universe, forcing the characters to join forces. It allows for a lot of interesting character interactions and power combinations, like Percy's water powers combined with Po's martial arts skills.