Yes, Paul's death is ironic, as it is the opposite of what he intended. He rode the rocking horse furiously to determine which real horses would win races so that he could earn his mother's love. Yet he dies without winning it.
Paul's death is a sad irony. He was too young and innocent to know that there was no amount of money he could win that would fill the hole in his mother's heart. No matter how much he won, she would always need more. He never would have been loved by her either, because she had no capacity to love her children, as the opening of the story informs us.
Paul took literally his mother's stated need for money. But it is clear she was using the money to try to fill in for some other lack in her soul. Paul was inevitably going to kill himself if he kept try to get her more of her drug of cash. She was like an addict—but he didn't understand that.