

Luigi and the Penguin King bonding
While Luigi, the Penguin King, and Lumalee were working together, after they they had a meal that Lumalee had for them to eat,
they had a heartfelt conversation and started to get to know each other more. (This is similar to the Fire Flower Field scene where Mario, Toad, and Princess Peach were bonding more.)
Luigi talks about him and Mario, about what they've been through, what he has been through
The Penguin King talks about that he'll do anything to save and protect his Kingdom, his people, and his family, Luigi knows that feeling, and so does Mario.
They're the same. They do everything for their home and their family. Then they'll shake hands, considering that they are friends.
After bonding, they got up and continued on forward.