Welcome, Sonny!
As Toni said, "legitimacy" is a big part of the thread of "politics" in GF. Michael provides a Grade-A example in the scene where he woos Kay in New Hampshire. He says his father is "just like any other powerful man with responsibility for others." Kay says "Senators and Governors don't have people killed," and Michael replies, "Now who's being naive." He's saying, "If the political pezzanovanti can lie, cheat, steal, kill, etc., and be considered 'legitimate,' so should I."
The other aspect of politics that applies to organized crime is license--that is, political protection that enables gangsters to operate rackets with immunity from arrest or prosecution. That's what Sollozzo needed from Vito.