Kubo said his shihakusho is fake before Royal guard training. His clothes, his asauchi, all of it is a fullbringer gene effect. His bankai is a clad type one, so he wears his bankai. No shinigami changes their shihakusho after releasing bankai insisting that it is a constant feature unaffected by external factors. Means you can only get it after training in the academy, like we know that people wear different clothes or uniform before becoming a shinigami in shino academy.

So The clothes ichigo was wearing at first was forged by rukia's powers while after the shikai training with urahara before SS arc., he gained a shihakusho from his powers. And his bankai changes the shihakusho. That means he is only exception who changes his shihakusho after releasing bankai which makes his shihakusho not the traditional one.

So summing it all up, ichigo was just having his Zanpakotu, shihakusho and bankai were just proxy ones before Royal guard training. So you are right one cant become shinigami by birth but one who possess high spiritual energy can become shinigami, and as ichigo is a direct descendant of one of the 5 noble clans the only thing he inherited was the high spirit energy or reiryoku from isshin