It's a strange question: should a 26-year old chimp named Hiasl qualify for human rights? It's strange because it's the wrong question. The issue is not whether Hiasl is human-like enough to qualify for such privileges, but whether or not he's person enough to earn such rights and protections.
The mistake is in making human DNA the metric for personhood. This smacks of human arrogance and exceptionalism; we are not the only persons on the planet -- far from.
Instead, what's needed in a situation like this is a list of personhood criteria so that an agent's level of personhood can be assessed. This will eventually have to be done for a number of reasons, whether it be to broaden the sphere of rights to include nonhuman animals, or to recognize personhood characteristics in artificial entities and technologically enhanced humans.
I've written on this topic in the past: