Side note: The Buddhist 'hamsa' in SE Asia derives from the 'hamsa' frequently mentioned in Ancient Indian texts. Oddly, it's not clear which bird the hamsa refers to, or if it even refered to different birds - perhaps a goose, perhaps a swan.
The Mute Swan overwinters in parts of North West India, and perhaps their range was more extensive in ancient times. This is the region where the Vedic corpus was written down, and in which the hamsa makes frequent appearances. In other parts of North India, the bar headed goose is more common. In ancient depictions of the hamsa in Southern India, it's more goose shaped. Ancient Buddhist murals from the Ajanta caves in Central India shows the hamsa as more swan like.
Some scholars have suggested that different references to the hamsa in the Vedic texts may refer to the goose in some places, and the swan at other places.
Fun fact: Many European languages have a word derived from the same root. It's χήν in Greek, Gans in German, ganso in Spanish, and goose in English.