Does bullying exist in the animal kingdom?

Rarely, but it can happen. African wild dogs tend to bully the weakest female. It can get to the point where they also bully her offspring.  There is a famous documentary made about this by a biologist who studied the wild dogs.  I saw it years ago, but I'm sorry, I don't recall the name.

Some horses bully babies.   My stallion loved his babies and when the babies had to be weaned, I put them in with their dad.  They picked up his wonderful personality and learned good habits.  However, this is rare.  Some stallions chase babies and are dangerous.

Goats tend to get along and usually look after each other.  My lead buck would always be the last one into the barn because he waited for the others and made sure everyone was safely inside before he went in.

Guinea pigs are very protective of their babies.  Most male guinea pigs, if raised properly, will take care of the babies once they are weaned from their mothers. I have a 2-month old male baby with 3 older males and you would think he has 3 mothers!  All my male guinea pigs have adored the babies.

Animals usually become mean to each other when they are improperly cared for.  Animals which are consistently underfed or starved can resort to cannibalism - even animals which are strict vegetarians.  Animals which are too crowded will fight for their territory.  Chickens are notorious for this, which is why factory farms remove the tip of the chickens beaks.  Such a terrible practice is not necessary on decent farms which either let the chickens free roam or have spacious pens.

Pigs, likewise, are very clean animals if kept in a pasture.  It's only in filthy barns and feedlots where pigs smell bad because they are kept in terrible conditions.

Finally, it's worth commenting on the ongoing hatred between lions and hyenas.   They simply dislike each other.  African biologists describe it as a war.  No one is really sure why they dislike each other so intensely.  One theory is that the hyenas provoke the lions by invading their territory and trying to steal fresh kill or prey on the lion cubs.

The first thing that comes to mind is chimpanzees! People don't realize this, but chimps are actually very mean towards each other, and even form gangs. These "gangs" will go around and attack other chimps that aren't in their "gang." Bonobos, on the other hand, look exactly like chimps (but are still a different species), but are nice and are not violent.