Kaido really did owe here that life long debt.
I wonder if Linlin was being sincere the whole time that she said she saw Kaido as a little brother. Did she perhaps have the feels for him too?
Kaido and Big Mom is very interesting.
A: Linlin loves other races/Ancient Races, look at her fangirl over King. Big Mom loves equality/racial diversity to the extent she tried to recruit her former Nakama/Little Brother's Top Subordinate to her crew.
Linlin liked Kaido BECAUSE he was an Oni.
B: Linlin is protective of younger people (see Tama) at times. I think since Kaido was the Cabin Boy/youngest, he got picked on, a lot. Linlin resented this, and protected him.
C: I think Kaido opened up to Linlin about his past before Vodka Kingdom, and his parents. Linlin is dismissive of her children, but not uncaring. If you hurt her children, she will hate you for it; Linlin took Pedro's eye for stabbing Tamago's eye out, when Tamago wasn't even her child. Linlin is know to be kindest to her youngest children: since Kaido was the youngest Rocks Pirate, she looked after him.
Kaido is the only individual we've seen in the series who Linlin treats like an equal.
Linlin just legitimately loves Kaido as family. Even if Kaido isn't Katakuri's Dad, Kaido being Kat's cool Uncle is also awesome.