Except, the ATLA series ISN'T western in it's culture.
It's only made IN the West.
In ATLA, you have a complete mix of different cultures and in a pseudo-medieval land like ATLA, surnames aren't very important. At least, the way they are today.
What was actually important was rank, like the fire lords / ladies, Toph's family, etc. and how much coin you carried.
If you had a last name, it was normally either from your occupation, where you came from (for example, Mako of Ba Sing Se and then a surname would trickle down from there), your father's given name and -sen / -son, or even something as simple as a word, like Lake.
Now, given the strong Chinese (Earth Kingdom) and Japanese (Asami Sato, the Fire Nation somewhat) cultures, THEIR last names could be trickled down from their own personal languages.
It just so happens that everyone in the Avatar-verse speak English.
OR, there are more continents than where the Gaang and Korra and Co. live.
There's a wealth of possibilities, but you need to look at points further than, "these people have last names, why not them?!"