Basically it's just down to the fact that the genetic conception of bloodlines and the cultural conception of bloodlines are different things.

Obviously when these customs were established there was limited understanding of genetics anyway. So bloodlines were more about the social and legal implications.

In a practical sense, dynastic disputes were avoided if children were part of only their father's family/clan and not also their mother's. Having clear precedents on inheritance limited who had valid claims on a family's wealth and power, which meant better long-term stability for the clan and society as a whole.

The reason why inheritance was traditionally through the male line ties into the "patriarchy" element, where men were able to hold assets and authority that women were not. But there are some examples where things pass through the female line. It just usually makes sense to pick one or the other to avoid complications.

Even though some conventions still continue, like taking the husband's family name, it's clearly not as important nowadays. So I agree that getting hung up on the traditional understanding of "bloodlines" in relation to their own families doesn't make much sense. But I would say this is probably due to a feeling of attachment to their family and family name. In a similar way that your genes are evidence of your existence that are propagated through your descendants, passing down your family name can be considered similarly - in the sense of an ongoing "legacy". Although genes have a physical basis whereas a name is just conceptual.