Yes, whales do have belly buttons.
Whales are marine mammals and like (almost) all mammals one of the characteristics they share in common with other mammals is giving birth.
The belly button is what is left after the baby whale losses its umbilical cord (aka birth cord) which is attached to the abdomen of the new-born baby whale.
During a whales gestation period (the period between conception and birth) the umbilical cord is connected to the placenta and the baby whales stomach.
This cord plays a very important role as it transfers oxygenated, nutrient rich blood to the fetus, while also transferring nutrient depleted blood back to the placenta and removing toxic waste.
This filtration system plays a major role in a fetuses development process and ensures the child is getting the nutrition it needs.
After birth occurs the umbilical cord is separated from the baby whale and mother allowing the baby whale to receive food externally.
In the beginning feeding generally involves the mother producing a thick paste like milk which the baby whale suckles from her nipple.
Depending on the species of whale a baby whale may continue to feed on its mothers supply of milk anywhere from 6 months to 2 years or until it is able to fully survive and hunt on its own without the need of its mothers nutrient rich milk.
Some children will continue to suckle way after the mother no longer produces milk and will not stop until they are able to psychologically separate from the mother and their dependency on her.
Eventually, the baby whale will grow into an adult and repeat the process with their mating partner.
As you can see the belly button of a whale or human is a reminder of the baby’s time within the mother’s womb and the role it played in providing the child with the nutrients it needed to survive.