What are you trying to use the Illustrator path for? Are you wanting to use it as a motion path? If so, you need to select the “Position” property and THEN paste the Illustrator path. You’ll end up with position keyframes that match the shape of the pasted path. Just pasting a path onto a layer will not work… unless…
Are you wanting to bring in an Illustrator path as a mask? Simply selecting a solid and then pasting an Illustrator path should result in a new Mask being applied to your solid using the Illustrator path as the shape of the Mask.
Not sure why your path doesn’t seem to be on your clipboard.