Chimpanzees, typically, kiss and embrace after fights. They first make eye contact from a distance to see the mood of the others. Then they approach and kiss and embrace.
Frans de WaalEmpathy as a complex emotion is different. It requires awareness of the other person's feelings and of one's own reactions. The appropriate reaction may not be to cry when another person cries, but to reassure them, or even to leave them alone.
Frans de Waal