Personhood does not exist. It is a social construct that human societies use to include and exclude people. Once, women weren’t persons. Now they are (to most of us). The same goes for people with skin of a different colour, people who worship relatively strange gods, and other similar “aliens”.
Societies start including a class of people into the folder they reserve for persons when their general knowledge base expands and causes them to look upon those people with empathy. When you see your own experiences reflected in someone else’s life, it is difficult to think of them as anything lesser than you. Think of any movie you have ever seen where two enemies bond over shared experiences of pain and happiness. You get my point.
In addition to this, we are the only species who bother with such classifications. A crow does not care if you are a person because crow society has no need of you. Dolphins — which may have recently been classified as non-human persons (or have they?)— can care about your well-being without putting a label on you.
We do language. That’s who we are. And in doing language, we come across category issues such as this one. The way out is realising that the category is made up, and we can always make up new ones if we like.