"Born and bred in the city-state of Ishgard, the majority of company chocobos are geldings"

Geldings are castrated males, usually a term for horses as well. It's in the actual chocobo description that we have unbreedable male chocobos. There are reasons for this in the cut scenes, mostly saying that the Ishgardians didn't trust the Grand Companies with females or breeding males as they felt that the population would get out of control (not so much in these words, but heavily implied) and they gave us geldings so that they would be more compliant.

Somehow Godbert twisted their leg however for the Gold Saucer.