It's not that the spirit would care about my bloodline, exactly --- more that my bloodline gives me the magical ability to both summon the spirit and ensure that it can't overpower me.
Admittedly, it's also a bit more complicated than blood, as well --- we mages collect the entire volume of our magical knowledge into a family crest that can be transferred from one person to another. This is technically possible to do between unrelated parties, but the recipient of the crest is much less likely to have their body reject it than an extremely close relation who's been prepared for it from a very young age, so that's where the blood aspect comes in.
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Admittedly, it's also a bit more complicated than blood, as well --- we mages collect the entire volume of our magical knowledge into a family crest that can be transferred from one person to another. This is technically possible to do between unrelated parties, but the recipient of the crest is much less likely to have their body reject it than an extremely close relation who's been prepared for it from a very young age, so that's where the blood aspect comes in.