The actual names of the two Curtis brothers are Ponyboy and Sodapop. When Pony meets Cherry Valance for the first time at the drive-ins, she asks him his name. Pony mentions that he hates telling people his real name and tells Cherry his name is "Ponyboy Curtis" (Hinton 20). Pony is pleasantly surprised when Cherry tells him that he has an original and lovely first name. Pony was probably used to getting made fun of or listening to sarcastic remarks when he told other people his real name. Ponyboy then mentions that his father was an original person. Pony's father must have been a fun, creative individual in order to give his children such unique names. Pony goes on to tell Cherry that he has an older brother named Sodapop. He then tells Cherry that it says Sodapop on his brother's birth certificate as confirmation.
These names are their real names, not nicknames. We know this because Ponyboy says that these are the names that are on their birth certificates.
We find this out in Chapter 2 when Ponyboy and Johnny are at the movies. They meet Cherry and her friend Marcia (the Soc girls). Cherry and Pony start talking. She comments, of course, on his name. He says that people usually think he's joking. He goes on to say that both his name and Soda's are real. He is happy that Cherry does not think he is joking -- she just says it's an original and nice name.
These are the actual names of the characters. In the book, Ponyboy defends his name by stating that it is the name that is seen on his birth certificate as well as Sodapop's. He also states that his parents had a sense of humor, and this is why they were given these names.
No, Sodapop and Ponyboy are not nicknames. In one of the scene in the movies Cherry ask if it was his original name and he answered that his father was an original man. So no it's not an nickname it is an, "original" name