Adobe After Effects trying to put an image on an animated shape layer

If you've already exported a PNG from your Illustrator document, choose File > Import to bring that into your After Effects project.  That can be linked to an existing Shape Layer if that's what you're looking to do with the Parent options for the resulting Layer when you add the PNG to a Composition.

You should be able to import your Illustrator file directly.  If all of the objects in the Illustrator file are outlines, you can add the Illustrator Source Footage Item to a Composition, select the resulting Layer, and then choose Layer > Create > Create Shapes from Vector Layer.  You can work with the resulting Shape Layer or copy and paste the contents to an existing Shape Layer.