King K. Rool is a crocodile and he has a navel, in real life crocodilians and other reptiles do not possess navels, this trait is only found in mammals, more specifically placental mammals. He's an anthropomorphic fictional character, not an anatomically and scientifically correct animal.
The underwater levels take place below the surface of the lakes on Donkey Kong Island, but have wildlife that only inhabit oceans. This isn't the real world, so in this fictional world it's possible they're creatures that can live and thrive in lakes. Additionally, it's possible there may be underwater passages (not seen in the levels) that connect the lakes to the ocean.
It is not possible for Donkey Kong and Diddy Kong to breathe underwater while scuba diving without oxygen tanks as they are mammals. If Donkey Kong and Diddy Kong tried this in real life, they would both drown. It's a video game, not real life. This is common in underwater levels in games.
The Kritters in variations that can jump, their gold-plated armor disappears when they do so, and reappears when they land.
When swapping places between both Kongs in play, only the high-pitched vocals typically heard when Diddy Kong is released from the DK Barrel or taking the lead is used. Normally, the low-pitched vocals would be used if releasing Donkey Kong from his barrel or putting him in the lead.
In one of the game's artworks, Diddy Kong is shown riding on Donkey Kong like the team-up ability that was not present during gameplay until the later two games in the DKC trilogy.
Although Rambi's bonus stage is set on a snow-covered mountain like the levels Snow Barrel Blast and Ice Age Alley, he does not appear in either levels of that setting.