Stray Italian Greyhound has the ability to refer to many things, including politics and romance. But the song was actually written before the presidential elections of 2008, in which President Obama ran.
The song was written around the time President Obama announced his presidential run. Vienna Teng thought, "Wow, this guy has really good ideas and seems to have a really good heart that will hopefully not be corrupted by politics." While being cynical would have been easier, Vienna thought, "I really want to help make this happen." .