Final answer:
Animals and plants are classified in the domain Eukarya in the Three Domain system of taxonomy because their cells have a membrane-bound nucleus.
Step-by-step explanation:
In the Three Domain system of taxonomy, living things are grouped into three categories, one of which includes the domain Eukarya. The most outstanding feature that most animals and plants share, setting them in the Eukarya domain, is the presence of eukaryotic cells which are characterized by a membrane-bound nucleus.
This contrasts with the prokaryotic cells of the Bacteria and Archaea domains, which lack a membrane-bound nucleus. Additionally, eukaryotic cells also contain other membrane-bound organelles, which further distinguishes them from prokaryotic cells. Thus, the correct answer to what aspect of cell structures most animals and plants share in common that sets them in the shared domain 'Eukarya' is A. cell structures defined by cell membranes.