The origins are in human nature. As you yourself note, blood is used in most languages to describe kinship. Biologically, we are born of the flesh of our parents. Their blood is said to flow in our veins and modern discoveries of genetics and DNA prove that.
Thought.co has an article about kinship and enumerates its types, of which the first is of course:
Consanguineal (of the same blood): This kinship is based on blood—or birth: the relationship between parents and children as well as siblings, says the Sociology Group. This is the most basic and universal type of kinship. Also known as a primary kinship, it involves people who are directly related.
Blood is what it is, and flesh is too. Indeed, in times of old flesh and seed were also used to describe lineage:
...concerning his Son Jesus Christ our Lord, which was made of the seed of David according to the flesh. (Rom. 1:3 KJV)
Blood is described as a means of transmitting filiation and identity in this article:
During the past several decades, historians of Rome have been considering a series of questions raised by anthropologists about the symbolic meanings of bodily humors, especially blood. It is becoming evident that in ancient Rome, blood—the fluid substance linked with filiation and the transmission of identity—was the object of a set of perceptions much more complex than used to be thought.
(Moreau, Philippe. “The Bilineal Transmission of Blood in Ancient Rome.” Blood and Kinship: Matter for Metaphor from Ancient Rome to the Present, edited by Christopher H. Johnson et al., 1st ed., Berghahn Books, 2013, pp. 40–60)
Here is an interesting thread which claims that the origin of the use of blood to describe kinship lies in Aristotle. This may be the answer you are looking for:
We talk about bloodlines and blood relatives because of Aristotle. The short version is that Aristotle believed that semen was a highly purified form of blood, and that menstrual blood was similar but less purified, and the blending of the two resulted in the conception of a child. Since much of Western philosophical and scientific thought centered on Aristotle for many centuries, some of his terminology has stuck long after science has moved past the ideas behind it.