In my opinion the lion is not asking for forgiveness. He is simply greeting the dog in a friendly manner. It's almost like handshaking and it practically mimics the handshake of humans (but not during the corona virus pandemic!). There may be an element of reinforcing friendship in the greeting which is what friendly greetings are all about anyway.
Immediately after the "handshake" the dog turns and leads the way. The lion follows. Perhaps the dog is the leader in this super duo? This is my interpretation. There is one thing certain: they are very close emotionally. They have formed an incredibly strong bond which clearly indicates that they were raised together and have lived side by side since they were toddlers to use language designed for people.
Emotions and self-awareness
The act of asking for forgiveness requires that the non-human animal or human animal (human) asking knows that they're done something wrong. The way that they know they have done something wrong is by measuring their behaviour against some standard or norm. Those standards and norms come from society in the human world. Can dogs and lions have their own standards and norms that relate to a friendship like this? I would doubt it.
Also the act of forgiveness probably also requires the ability to be self-aware. You have to be able to look at yourself from outside yourself, objectively. It's as if you are measuring your behaviour against some standard and this requires self-awareness. There are doubts, considerable doubts, as to whether cats can be self-aware. There are also doubts about the higher emotions in domestic and wild cats. Forgiveness is born out of a feeling of guilt and perhaps shame. These are higher emotions. I would doubt that the lion feels these emotions. This is not to in any way denigrate this beautiful relationship and the tender behaviour of this fantastic looking lion. I'm just trying to look at it realistically.
It is well known that cats make friends with other cats and have interspecies friendships with, for example, dogs. Cats have friendly greetings like the tail-up position and the nose touch. Friendship is based upon affection and affection is an emotion which most people agree cats can experience. Of course dogs make friends with other dogs and their owners as well. These thoughts support my assessment that what we see in the video is a very friendly greeting which may have been trained into the lion and the dog by their owner. We don't know.