Even Christian saints can fly, like shamans. Many people outside shamans believe in the bridge mythology surrounding the journey to the underworld. There are rituals designed to build this symbolic bridge. It is like a rite of passage where the shaman proves his worthiness. Ladders are used to get to the heavens, and for gods to come down to earth.
Many cultures use ladder terminology to describe the ascent of dead bodies.
Shamans are not alone in their powers- leaders of all religions have similar powers. There are parallels to symbols used, ideas, rites, etc. There are certainly some Sino-Buddhist influences in Tungus shamanism, but also Indian and tribal influences played an important part. Shamanistic practices can be traces back to Paleolithic hunters. Shamanism has grown, changed, and adapted as time went on, as other areas and religions came in contact with it and influenced it.