Is there a way to determine whether an aromatic compound originates from the polyketide or shikimate pathway by looking at its structure? If so, how?
If both pathways produce the same molecule, you can't tell: It's the same molecule.
You could knock out some enzymes on either side and see which ones affect output. Or you could spike the cells with a radiolabeled precursor of only one pathway, and then see if the output is also labeled.
However, you say "by looking at its structure", so I'm assuming no additional experiment can be done - then no, except for the trivial case where only one of the pathways produces the molecule (in which case it obviously came from that one).