The option was "Comments" before. Then I switched it to "Discussion board".

The available comments disappeared. They can only be found again if I switch back to "Comments".

So the question is can I somehow move existing comments into the discussion board?

No, it's not possible. That's been on leafo's wishlist for a long time, but was never implemented.

At least the comments do reappear, if you do not like the discussion board style.