In the manga it's mentioned that Zeldris has the power of the demong king. And I believe in the anime too. What does this mean? Why Zeldris has the power of the demong king?
Because he's basically Demon King's right hand
In the Drole and Gloxinia's flashbacks, already after Meliodas's treachery, Zeldris is a) Meliodas's successor as the only other Demon King's child b) supposedly the new Ten Commandments leader and b) "fixer" as mentioned by Gowther, when Demon King realizes that Gowther got out of his prison and Zeldris was sent to get him back. The Demon King's power that Zeldris has is simply granted by Demon King for Zeldris to do his bidding