People already have an ID in the United States, how to apply for a green card in the United States?

The key is how your mother applied for you to come. It is impossible to have SSN ID card and green card.

Did you give you a number starting with a when you went through customs? If you have it, that's your green card number. In your case, I think your mother applied for a green card through political asylum. In this case, you have to live for two years to get it, and you can't leave the United States for two years.

So there is no need to reapply at all. You can apply for this green card by passing the medical examination and sending your information. You'd better ask a lawyer to help you, because you need to fill out a lot of documents and forms. This doesn't need your mother's guarantee, but it needs relevant proof of your previous relationship, birth certificate and so on. Find a company to help you. There are many such companies, and they don't need to spend much money.