Is it possible that the baby was switched in the hospital given the difference in his physical characteristics (facial features, shape of fingers, eyebrows, skin color, hair, nails) from all members of his family, and is it permissible for him to undergo a DNA test to confirm this?

Denying resemblance does not necessitate denying lineage. As stated in the Hadith, when a man had a black child born to him, the Prophet, peace be upon him, affirmed the child's lineage to him despite the difference in color, reasoning that camels might give birth to offspring of a different color than their parents due to a genetic trait. Ibn Hajar transmitted from Ibn al-Arabi that a child is attributed to his father even if his color differs from his mother's. The hypothesis of substitution in the hospital is not considered, and DNA testing is not valid for denying lineage, though it may be used to establish lineage when needed.