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.

For a somewhat longer version, let's go straight to the source. Aristotle discusses the nature of semen at some length in his Generation of Animals, which you can read here. He ultimately comes to this conclusion:

We have previously stated that the final nutriment is the blood in the sanguinea and the analogous fluid in the other animals. Since the semen is also a secretion of the nutriment, and that in its final stage, it follows that it will be either (1) blood or that which is analogous to blood, or (2) something formed from this. But since it is from the blood, when concocted and somehow divided up, that each part of the body is made, and since the semen if properly concocted is quite of a different character from the blood when it is separated from it, but if not properly concocted has been known in some cases to issue in a bloody condition if one forces oneself too often to coition, therefore it is plain that semen will be a secretion of the nutriment when reduced to blood, being that which is finally distributed to the parts of the body. And this is the reason why it has so great power, for the loss of the pure and healthy blood is an exhausting thing; for this reason also it is natural that the offspring should resemble the parents, for that which goes to all the parts of the body resembles that which is left over. So that the semen which is to form the hand or the face or the whole animal is already the hand or face or whole animal undifferentiated, and what each of them is actually such is the semen potentially, either in virtue of its own mass or because it has a certain power in itself.

Yeah, Aristotle isn't exactly known for his easily approachable writing style. The gist is that semen is blood, and that it contains the essence of all parts of the body within it. This explains how something so small and seemingly insignificant is nonetheless able to give rise to an entire person.

But Aristotle recognized that children have features from their mothers as well - they aren't simply images of their fathers. Menstrual blood is analogous to semen, in Aristotle's mind:

It is plain, then, that the catamenia [menses] are a secretion, and that they are analogous in females to the semen in males. The circumstances connected with them are evidence that this view is correct. For the semen begins to appear in males and to be emitted at the same time of life that the catamenia begin to flow in females, and that they change their voice and their breasts begin to develop. So, too, in the decline of life the generative power fails in the one sex and the catamenia in the other.

A long discussion and further explanation follows. But to Aristotle the menstrual fluid, while analogous, is not equivalent to semen:

For the catamenia are semen not in a pure state but in need of working up, as in the formation of fruits the nutriment is present, when it is not yet sifted thoroughly, but needs working up to purify it. Thus the catamenia cause generation mixture with the semen, as this impure nutriment in plants is nutritious when mixed with pure nutriment.

The overall effect, Aristotle summarizes, is that

what the male contributes to generation is the form and the efficient cause, while the female contributes the material.

In other words, the semen/blood from the man contains the form of the child, and the menses/blood from the woman contribute the actual materials. He later gives an analogy:

Just so no material part comes from the carpenter to the material, i.e. the wood in which he works, nor does any part of the carpenter's art exist within what he makes, but the shape and the form are imparted from him to the material by means of the motion he sets up. It is his hands that move his tools, his tools that move the material; it is his knowledge of his art, and his soul, in which is the form, that moves his hands or any other part of him with a motion of some definite kind, a motion varying with the varying nature of the object made.

So, to Aristotle, children are literally made from the blood of their parents. The mother's blood provides the physical materials for building the baby's body, while the father's blood/semen is essentially the directions for putting it all together.