In Beckett's Endgame, Hamm has a toy dog that, for most of the play at least, he cherishes as a companion. Even as the world hurtles towards an apocalypse, old habits die hard, and Hamm is incapable of showing affection toward human beings or other animate objects. He has his toy dog, and that seems to be enough.
On one level, the existential level, the toy dog and the affection that Hamm lavishes upon it could be seen to represent the way in which humans are fundamentally alienated from each other. As they cannot properly connect, they resort to creating objects with which they can form the ties that they really should be forging with their fellow humans.
At some level, Clov realizes this, which is why he's so jealous of Hamm's toy dog. The stuffed animal is taking his place in the long chain of intersubjective connections that holds everyone together.
In due course, Hamm will reject the toy dog and throw it away, a sign that even an artificial connection with a synthetic creature is too much for him. In doing so, he isolates himself further and prepares to face the apocalypse alone, in true existentialist fashion, without any fear or illusions.