Question

how was it known that ashoka and piyadassi was same person

Answer 1

because of ashoka's friend who told every thing about him

Answer 2

In various rock and pillar edicts established by ashoka, he was described as devanampriya or the beloved of the god. After the kalinga war, the rock edict he set up, mentioned him as piyadasi,(piyadassi or priyadarshini) or the seer of the the one who looks with kindness upon everything.