LOL on the "play one on Wikipedia" comment. "An office under the United States" is, AFAIK, any government office, federal, state or local. I would imagine that, just as it would be a breach of separation of powers for someone to be simultaneously Congressman and President, it would be a breach of federalism for someone to simultaneously serve in two different governments. I have no idea why the seniority date isn't the actual date of oath-taking, but he couldn't have voted on federal legislation until he resigned as a city councilman and was sworn into Congress, so there was, in reality, no dual office-holding.