In addition to Israel being God's firstborn, and also the wages of sin being death and Jesus being both firstborn that dies due to (our) sin, and the lamb's blood smeared over the household (God's household) so that we are passed over.
In answer to your other question...
Did the firstborn hold a privileged status?
...the answer is YES, in all caps and bolded.
Through most of history, in most of the world, firstborns were given preferential treatment. The firstborn has greater authority within the family, and a much larger inheritance, if not the entirety of the inheritance (though small inheritance gifts might be given younger sons). The firstborn is seen as the continuation of the family line, and as much aid is given them as possible in the hope that one day the family may become powerful/successful/secure.
This idea is called Primogeniture, and is a huge deal in many cultures, ancient and modern.
Children are the "social security" of families, and special emphasis is usually put on the firstborn son as the hope and security of the family. Not just greater privilege, but also greater responsibility. For example, it was Jesus' responsibility, as the firstborn of Mary and Joseph, to care for them in their old age. This is why Jesus' last act was to ensure His mother was taken care of, as was His responsibility as the primogenitor (John 19:26-27). Jesus' brothers also came to take authority from Him, thinking Him mad and thus unfit to lead the family (Mark 3:21).
In Ancient Egypt, by law, firstborns where given double-portions of inheritance, and became the head of that family's household, and inherited governmental offices - think European ranks of Lords and Dukes being inherited. (Oddly, inheritance of the rank of Pharaoh works uniquely differently). This may seem quaint to modern Westerners, but the importance of family lineages and households are a big freaking deal. Many families today go without, sacrificing for their children's future. Primogeniture is one of the key ways that used to (and still does) occur.
It's a big deal in ancient (and sometimes modern) Judaism too. (See also Deuteronomy 21:15-17, demanding that you can't play favoritism and overlook one wife's firstborn in favor of another's).
This is why is was a huge deal for Esau, the firstborn, to sell his birthright to Jacob, who was the second. And also why God made a point of saying, Jacob He loved, but Esau He hated - because birthrights and social ranks aren't of value to God, rather the posture of the heart.
Same with King David, God looked past Jesse's many other sons, and looked to the youngest (David) to anoint and make king, because God was interested in David's heart (calling him, "a man after My own heart"), as David regularly worshiped God in the fields as a young shepherd. (King David also seemingly went against Deut 21:15-17, by making Solomon (firstborn of his favorite wife, Bathsheba) king instead of his many other children by other wives).