Chaos offers world collision on cloth.
Apex/physx does not.
Know what you are using.
Also:
A cape should likely not be fully simulated anyway.
You can set where it should be by giving it a properly weight painted and styled animation. With that, you can apply a light distance value to the cape to get the cloth effect going.
Normally, nothing on the character interacts with anything else.
To get things working with the horse, you would have to join up the clothed character to the horse in a single asset - then you can get the self collision to work on a fully simulated cloth even in PhysX.
Runtime character merge (used for modular characters) could be a good way to achieve this…