Yes, they are both beef.
The word "hamburger" can refer to the ground meat, (ex: "I have 2 pounds of fresh hamburger.") and it can refer to the sandwich made out of a cooked patty made from hamburger meat. (ex: "I ate a hamburger for lunch.")
Cheeseburger is just a hamburger sandwich with cheese added to it. The word cheeseburger does not refer to any one ingredient - only to the whole sandwich.
There are vegetarian hamburger substitutes made from soy and vegetables which might be appropriate for you, such as Boca Burgers.