No one else will say it, so I will
Linlin worrying about Kaido, who used to be like a little brother to her, is why I love Big Mom.
Even as crazy as she is, her "Mama mode" is so strong that she can't bear to see Kaido get hurt.
To Linlin, Kaido will never be a Yonkou. He's the awkward, angry kid she met decades ago, and it pains her to see him hurt. Same way Kaido cried when he realized Linlin lost to Law and Kidd.
Kaido was wrong, though: Enma drains Haki, then releases it. If Oden's Haki ever tested Zoro, it's when he first received Enma in Wano from Sukiyaki.
In Enma releases Haki that easily, it makes no sense that the blade is full of Haki. I think Kaido was confused because Enma is THAT GOOD at releasing Haki.
Oden scarred Kaido, if it was Oden's Haki, Kaido would know. That's why he was confused: Kaido knew how Enma produced Haki, but Zoro's Haki is different than Oden's, so Kaido didn't know what was going on.
"Ahhh, that hurts! I see... That must be Oden's Sword!"
Oden's Sword, not Oden's Haki.
Kaido realized in this moment that Enma was Oden's Sword, and that's why he was familiar with Enma's Haki flow: we know from Cursed Blades, like the Sandai, that some swords unleash more power naturally. They're not cursed, they're... hungry? Kaido recognized HOW Enma let out Haki, not Oden's Haki.