I don't think you can use an object for "take after" apart from the person taken after.
"In his looks, he takes after his father". To me (and don't take this as authoritative) it means something like "He takes (from the genetic pool) after (in the manner of, following) his father."
or "He takes ..." in the manner that a liver transplant or a plant cutting "takes".
I find yours very interesting.
If we think about the original meaning of 'take', the explanation of 'take after' would be like yours.
I'm not really sure but...you may be right.
Take after is casual language which often shows more regional variation than more formal language. Here's an entry from a dictionary: take after to be like (someone, especially a parent or relation) in appearance or character
Example: She takes after her father.
And some examples:
“He takes after his father. All the men in his family are big.” "Jenny takes after her grandfather. They’re both really good with numbers."