Remember when Oda trapped readers with Bartolomeo's portrayal before it's revealed that he is Luffy fanboy?
He was portrayed to be an antagonist. His epithet is iirc Man Eater, not a friendly epithet and suggest he has killed many men before.
He reacted strongly upon hearing the word "Straw Hat Luffy".
And then it was revealed he is a fanboy.
I get the feeling something similar will happen to Who's Who. Oda hinted that he has some beef with Queen, wants to take out each other. While actually never confirming who they actually want to take out.
So they may actually not target each other, but both of them actually target another but same person. And it's X Drake. Not sure about their motives for targetting X Drake, because Queen doesn't seem to be the genius type to tell who's the mole.
But for Who's Who's motive, he lead a pirate crew prior joining Beast Pirates and X Drake was a Marine prior his acting as pirate career. So in the past, probably some Marines that included X Drake defeated Who's Who's original crew, thus he wants a protection from Yonko Kaido and had a revenge towards Marine and X Drake, the latter who somehow happens to join the same Yonko crew with him.