D. H. Lawrence's "The Rocking-Horse Winner" is centered around themes of materialism and greed, and in this respect, the image of Paul riding the rocking horse serves as a powerful expression of the indictment this story makes against materialistic culture and priorities.
It is noteworthy that your question itself describes Paul's riding of the rocking horse as "furious," which is an appropriate descriptor to use. Paul's riding is marked and defined by a combination of desperation and futility, as he chases after a goal that his actions can never attain. As his words to his uncle suggest, he is attempting to satiate his mother's greed (and perhaps, in this, to earn her love and approval), but his actions only spur on new expenditures, intensifying the very problem he was trying to alleviate. And yet, by this point, he is caught in the same trap that his mother is, with so much of his focus and energies directed toward the next race and the winning of more money.
There is a self-perpetuating quality to materialism where the pursuit of money and possessions becomes a goal in and of itself (independent of what value and satisfaction a person actually can attain from these things), and this fundamental irrationality is reflected in Lawrence's own description of the boy on the rocking horse. In Lawrence's words, Paul is "maddened," and even as he's dying, still his thoughts and energies are directed at the identity of the race horse, a quality that reflects materialism's all-consuming nature.