I think your question is slightly confused - the horse itself, as an inanimate object, does not have any feelings, but it definitely does produce feelings of anxiety in us as readers and in the other characters in this masterful short story. Think about the description of Paul as he rides on his rocking horse, looking for "luck":
When the two girls were playing dolls in the nursery, he would sit on his big rocking horse, charging madly into space, with a frenzy that made the little girls peer at him uneasily. Wildly the horse careered, the waving dark hair of the boy tossed, his eyes had a strage glare in them. The little girls dared not speak to him.
Such descriptions make us anxious for Paul because of the effect that riding on his rocking horse is having on him. Notice how this anxiety continues to be conveyed, with the concern of other characters at his state:
He would speak to nobody when he was in full tilt. His mother watched him with an anxious expressin on her face.
Even his self-absorbed and greedy mother is concerned at his intensity, though she can't understand what is happening. However, it is the last ride that Paul takes on his rocking horse that is by far the most disturbing:
"It's Malabar!" he screamed in a powerful, strange voice. "It's Malabar!"
His eyes blazed at her for one strange and seneless second, as he ceased urging his wooden horse. Then he fell with a crash to the ground, and she, all her tormented motherhood flooding upon her, rushed to gather him up.
The description of the sound of his voice and the way his eyes "blazed" suggests that somehow Paul is being possessed by some kind of spirit that comes from the horse, and obviously causes his mother to wonder and panic over what has happened.