Jacob received the birthright from Esau. It was not transferred by Isaac, and I'm not sure Isaac even knew about it. Rebekah and Jacob deceived Isaac to get the blessing, Rebekah having more deceit in her heart than Jacob did. We see later that even though many years had passed since Esau sold his birthright, he was still foolish (and hence would not be the best choice for progenitor of the people God had chosen for himself). Esau probably had become accustomed to being the firstborn with the authority and power that came with it—basically the equivalent of Isaac's authority and power.
Even though they deceived Isaac, they did not deceive God who knew the birthright had been sold. However, from Esau's point of view who still thought the blessing should be his, it looked like a theft of the blessing to him. The blessing was the passing on of Isaac's spiritual inheritance from God (Genesis 27:28-29). This wasn't necessarily the mechanics of how God wanted Jacob to become the Father of the Jews, but that method was the right way to do it in the hearts of all the people involved, and therefore was real as long as God was willing to go along with it.
This was a specific tribal culture and the way they did some things might not seem right to us without actually having been immoral to them. I don't know their culture well enough to say. But that would have affected how God interacted with them. (He treats us as we treat others.) Note this was before the Law was given.