For what it's worth, I once heard a sermon in which the preacher said it was not the same fellow.
My bible has the following intro to Obadiah:
Obadiah wrote this shortest book of the Old Testament probably soon
after 586 B.C. when the armies of Babylon destroyed Jerusalem. During
this conquest, the people of Edom helped capture fleeing Israelites
and turn them over to the Babylonians. They even took up residence in
some Judean villages. This angered the Lord, for the Edomites, as
descendants of Esau, were related to the Israelites (Gen. 25:21-26,
30) and therefore should have helped them. Obadiah prophesied that
Edom would be repaid for mitreating God's people. Within a hundred
years, Edom was in ruins. Obadiah also asserted that God is sovereign
over the nations and that the house of Jacob would be restored because
of God's covenant love for his people.
This would seem to preclude the Obadiahs being the same, as Elijah was well before the Judean exile (he was even before the northern kingdom was overrun).