Translation:Our last name is Wang.
I thought the same thing at first. After some research I found out that 姓 is actually a verb meaning 'to be surnamed' or 'to have the surname' so what you're really saying here is "We are surnamed Wang". So in this case, you don't need 的 because you don't really want to say "Our are surnamed Wang".
I think it is more logical to translate it as 'my family name is ' or 'my surname is' rather than 'my first name is' since we are all aware that when Chiinese give their full name it comes first with the given name to follow.