I landed on the island where SSS dlc takes place.

Entered the arena by climbing the mountain (or cliff) with the stone.

Talked to the merchant who sells potions and he talked about a device that I have to wear and collect the soul essence of the kith enemies I kill.

 

But I looked into my inventory after the conversation and I got no new items. Even the combat dialogue box did not show any new items added.

 

Is this a bug?

You will get it later after doing a few fights (to prove you are "worthy").

If you talked to him right after entering (and not signing up to be a contender), your response to him is "I will come back once I sign up for the fight". I came to him after initial fight in the arena. You might be able to go to him even before that, after talking to the caretaker of the arena.