What are some similarities between current American society and ancient Greece with respect to art, architecture, and politics?
Much of western civilization developed from ancient Graeco-Roman traditions. Thus, the similarities one finds between elements of contemporary North American culture and ancient Greek culture are actually the result of direct influence.
First, the notion of democracy originated in ancient Athens. Although the United States is technically a republic (or "representative democracy") rather than a direct democracy, many elements of contemporary government, including trial by jury, derive from ancient models.
In architecture, many buildings follow neoclassical models using elements such as columns of the three classical orders (Doric, Ionic, and Corinthian). The style known as "Federal architecture" is based on classical Greek models and can be seen in many iconic buildings such as the Lincoln Memorial, the National Gallery, and the American Museum of Natural History.
Another similarity to ancient Greece is the notion of public art as a form of civic pride. Just as Pericles engaged in a massive public works program to glorify Athens, so we now have requirements that a small percentage of the budget for public buildings be set aside for art.
In art, many of the techniques for portraying the human body, such as the S curve body shape, first appeared in Greek sculpture. The lost-wax casting process used by many Greek sculptors is still used in modern art.