My mother is dying, with her permission am I allowed to transfer her car to my name or should it be in both names?

So sorry to read about your Mother. May I ask if your Mother is self paid at the nursing home, or is she on Medicaid [different from Medicare] where Medicaid is paying for all of her care?

How can you demonstrate that your Mom is sufficiently cognizant to sign such a document. Even if Medicaid isn't involved....her estate may challenge the legality of such a "gift" in the last days.

Does mother have a will? Does she designate to whom the car should be given?

Permission is not good enough. She has to sign the title to transfer it. Better you find the Will and see what that says. And find a lawyer to read all the fine print.