Sometimes, if you're playing as Luigi and he falls off the track, he will say, "Thank you!" to the Lakitu who rescues him.
In Double Dash!! When you finish the race, the two characters you chose celebrate together. While this leads to the obvious niceness with pairs like Mario and Luigi or Mario and Peach, it becomes really sweet to see Donkey Kong and Peach or Mario and Bowser so pleased with each other. Especially during the award ceremony.
In the section right before the cave on Double Dash's Sherbet Land, there are two Shy Guys skating together. It's a small, but sweet sight to behold