People bring up the civil rights movement because it is the one of, if not THE main example of a people or group rising up against their oppressors. It inspired people internationally (study the Australia Black panther movement for example), so when people think of a movement to help defend people's rights, world-wide, they think of the civil rights movement, which due to unfortunate circumstances has been labeled a movement specifically for black people when it was anything but.
The comparison is made because many black people are homophobic. People make a comparison between civil rights movement and gay rights because denying gays rights and respect while being black is among the biggest form of cognitive dissonance imaginable.