President Barak Obama of the U.S., and Herman Van Rompuy and José Barroso of the E.U. held a news conference following the EU-US Summit at Lisbon in 2010. Even though the E.U. and U.S. are both empire-scale federal unions of states, and thus are equivalent in terms of political type or genre, they differ in terms of how their respective federal offices are arranged and constituted. This does not, however, nullify the basis of comparison.
For example, the E.U. does not have one definitive president; rather, the Commission has a president, as does the European Council. Interestingly, the alternative of having more than one president (e.g., a presidential council) was debated in the U.S. Constitutional Convention in 1787. Yet this difference does not mean that the E.U. and U.S. are not comparable. In fact, I submit that each union can be strengthened by such a comparison.
The picture below also shows the full presidential equivalence, while showing that the EU is not structured at the federal level as a replica of the US.
The placement of the respective flags is interesting because the
arrangement was, at least as of 2019, only done for countries. By implication,
I submit, the EU flag, rather than the flag of an EU state (e.g., France)
should be set with the US flag as a backdrop. At the very least, the EU flag
and the flag of the state of France should both be present, for the sovereignty
of the state of France is partial, just as is that of the state of Texas. Both
states, in other words, are in federal systems in which governmental sovereignty is split between
the federal and state polities.
Besides in how they are arranged, the flags themselves are also
visually equivalent. Both contain stars, which represent semi-sovereign
states, or polities, which are on the scale of the early-modern nation-state.
The early US flag is even more equivalent, for the number of stars is roughly
the same.
The E.U. flag was designed by Arsène Heitz and Paul Lévy for the Council of Europe in 1955. The EC and the E.U. would subsequently adopt the flag as their own. The "Betsy Ross" U.S. flag was in use by 1777. Regarding the number of stars, whereas the 13 on the U.S. flag represent the original 13 states in the U.S. alliance, the 12 stars on the E.U. flag were chosen when the Council of Europe had 15 members (10 of which were founding members). Even so, the similarity between the two flags is striking, especially as both the U.S. and E.U. have added states since their respective foundings. It would seem, in other words, that both unions have followed the same evolution, which is another signifier that the unions may be in the same political genre.
Lastly, the "EU and US" sign in the news-conference photo above implies that the two unions are equivalent. Lest it be objected, at least as of 2019, that the EU is not a political union, I submit that having a parliament, executive (the Commission), and supreme court (the ECJ) renders the federal level not only political, but also a government. The denial (in the E.U.) and ignorance (in the U.S.) on this point boggles the mind. Such is the force of ideology on cognition and perception.
The picture below also
shows the full presidential equivalence, while showing that the EU is not
structured at the federal level as a replica of the US. The placement of the
respective flags is interesting because the arrangement was, at least as of
2019, only done for countries. By implication, I submit, the EU flag, rather than
the flag of an EU state (e.g., France) should be set with the US flag as a
backdrop. At the very least, the EU flag and the flag of the state of France
should both be present, for the sovereignty of the state of France is partial,
just as is that of the state of Texas. Both states, in other words, are in
federal systems in which governmental sovereignty is split between the federal and state polities.
Interestingly, the photo below attests that U.S. flag flew next to the E.U. flag in front of an E.U. building in Brussels in 2012. Perhaps knowledge of the proper equivalence existed within the E.U.'s federal government, while still noticeably absent at the state level, including the residents thereof. 
Even though the E.U.'s federal level has been criticized for being populated by technocratic officials, I submit that the nature of the E.U.'s laws (e.g., directives, regulations, etc) are such whereas the officials have tended to have a good understanding of the E.U. as a political system meant in part to act as an international counter to U.S. power.
For more on this topic, please see: Essays on Two Federal Empires: Comparing the E.U. and U.S., available at Amazon.
