The fixation on bloodlines is often an artifact of the fixation of real aristocracies the world over. And there's a few different reasons for that. One is the old 'divine right of kings'; that some higher power put the royal family in charge to rule, and the descending tiers of nobility to serve them and administrate the lower classes. For adherents to that belief those anointed bloodlines are inherently superior in their roles and to disrupt that status quo is to go against both the natural order and the will of the divine. In our world it's bullshit made up by the people in power to justify retaining their positions, if they were just the scions of a family who happened to end up in charge in the distant past then it'd be no big deal to replace them, after all.
But in a fantasy setting with magic and spirits and possibly actual gods, the whole dynamic can have a bit more weight. Many of these sorts of settings have powers unique to the royal family or high nobility, and these are often in some way essential to the continued existence or prosperity of the lands in question as a way to justify that it actually matters.
But even when it doesn't, really it still goes back to imitating real world trends. Once it is established that there are separate social strata, how do they delineate them? It can't just be whoever has money and privilege at the moment, that's too mercurial. So, since inheritance rights also tie into bloodlines, we get the notion that it is indeed the blood which is what matters: X family descends from Y notable figure of the past, and thus is supposed to be in an elevated position; the children of a duke can lay claim to the lands and offices he held because they are of his blood. Some rando cannot, generally step in and take over without a whole bunch of caveats to make it acceptable.
And once we get into bloodlines being that important, you get families fixating on the 'purity' of that noble blood for its on sake. The traditional means of sealing a permanent alliance between powerful families was to marry, so that afterward you're all family and any children would have rights to both sides. But it's also leaning on nearly universal taboos against killing ones own family to try and reduce the odds of betrayal and murder, to varied success. Half the reason bastards were poorly regarded was the cultural institution of marriage being undercut by the production of children outside of it, but the more relevant side of it was that it muddied the waters on the issue of bloodlines as much of the time the couplings that resulted in them were inter-class.
In summary: bloodlines are viewed as important because they are the sole inflexible factor that the aristocracy can hold up as delineation between themselves and the regular people and are often tied into justifying the power dynamic in the first place. Adoption could happen, but the institution as a whole was leery of it since fiat declaration of noble worth sets precedent that they could be readily replaced, and indeed adversaries would be quick to declare such a child unfit or unworthy to stand among them.
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