The President and her brother
The sequence showing the Presidential debate in the first episode makes it clear that the election that was approaching was the general, not part of the primary campaign. It was stated earlier that the brother was the Republican candidate, indicating that the current President, the VP who succeeded Sheridan on her assassination, must have been a Democrat. So this must result in one of three highly improbable possibilities: either President Sheridan was a Republican who selected a Democrat as her running-mate, or Sheridan and her brother were in different parties, or the brother switched parties after the assassination and then immediately secured the Republican nomination for President.
So which is it, and in what possible universe does it make sense? It was also stated that President Sheridan was a dove whereas her brother was a hawk, which would seem to lend some credence to the second possibility, but two siblings existing as high-profile politicians in different parties is unlikely enough even before we consider that their father had also been a US senator.
Can anyone think of any real-life political dynasties involving family members in different parties? The closest I can think of is the Roosevelts - TR was a Republican and FDR a Democrat, but they were of different generations, and their relationship several times removed.