This matters because he is not saying he is a generic son. He is saying he is descended from one of Ghengis Khan sons with his first and supreme wife, Börte.
Börte was with Ghengis from the beginning. They were betrothed at a young age and married when she was 17. She was kidnapped early in their marriage, an event which the Secret History of the Mongols (a private family history document of Ghengis Khan’s descendants) as something that made her husband’s “breast torn apart” and generally was extremely upsetting.** Those feelings, and the fact that they and other information about her were considered something to record in the family history, suggests that Börte was important from the beginning.
He then rescued her, an event which marks the beginning of his many actions in The Secret History. Once he was powerful, she ran his court/the home camp (he was not home a lot while building his empire), and was essentially his empress. She was very powerful.
It is from her sons (minus Jochi, whose parentage was always questioned due to the time of her pregnancy and her period of capture*), that “legitimate” descent stems. For centuries in Central Asia and beyond, a ruler’s legitimacy stemmed at least in part from Ghengisid descent from these brothers. Leaders who were not part of the family, such a Amir Timur/Tamerlane, married into it to cement their own claims to power.
The Mongols did not practice primogeniture- great khans were chosen, sometimes by major leaders, but sometimes even by an entire clan (according the The Secret History, Ghengis Khan saw that himself when he was young Temujin, after his father died and all of his followers abandon his family rather than be led by his young children or his widow). Once Temujin was the great Ghengis Khan, then they had to be chosen from among his heirs, but any of Ghengis Khan’s sons by his most powerful wife were equally legitimate heirs, and their sons too etc.
*Unless that isn’t true, his paternity was not an issue, and the reason that Jochi is not part of the line of descent is simply because he died early and was not part of later events that determined who was a Ghengisid.
**The story of her kidnapping, as written in the Secret History, is quite moving. There were four people but only three horses, and the attackers were gaining. One person had to be left behind. As the others were men who might be killed, and as the attackers were motivated by revenge for the previous kidnapping of a woman from their clan, who then became Temujin’s mother, Börte was the “safest” to be left behind. Before Ghengis had to leave her, she gave him her shirt to remember her smell and remember her, and then he had to flee.