I've thought about this before and I suspect the answer is as simple as "it's one way".
Dumbledores explanation is as follows.
“But if Voldemort used the Killing Curse,” Harry started again, “and
nobody died for me this time – how can I be alive?”
“I think you
know,” said Dumbledore. “Think back. Remember what he did, in his
ignorance, in his greed and his cruelty.”
snip for brevity
“He took my blood,”
“Precisely!” said Dumbledore. “He took your blood and
rebuilt his living body with it! Your blood in his veins, Harry,
Lily’s protection inside both of you! He tethered you to life while
It's explicitly stated it's still Lily's protection magic at work here. Voldemort carries it too, but this doesn't mean that he is protected by it (note that Dumbledore does not say "He tethered you both to life"). From this we can assume the magic doesn't protect whoever has the blood, but the person for whom the sacrifice was made. Other carriers just act as neat backups for the original protectee, as it were.