The ordinary meaning of 独立 in Chinese is independence or stand-alone. After checking Japanese dictionaries, it seems that 独立 read as Japanese has a very similar meaning, albeit with different pronunciations.

I don't think you can split off 独立 from 金鸡独立 without changing the meaning. 金鸡独立 refers to the posture of standing on one leg, with the other leg lifted up, often seen in Chinese martial arts. Evidently, the name came from the imagery of chickens adopting this posture.