In Naruto, a number of characters don't have surnames, though many do. For example, Naruto is Uzumaki Naruto, while Gaara is just Gaara or Gaara of the Sand. Why is this?
There are a lot of speculations regarding this like on this Naruto forum.
- They only need a first name for story purposes, the last name doesn't matter so often.
- Either they don't belong to a known clan (or just don't have one) - making it unnecessary.
- Maybe its just the way Kishimoto planned it to be or to keep us in the dark or they weren't necessary.
- They don't have real names.
- Kishimoto just can't came up with last names to use.
All of these are speculations and opinions so there is really no definite answer for this.
In general for anime, according to TVTropes,
A major character is never referred to by their actual or full name, instead being addressed by a title, nickname, or Only One Name. Reasons for this vary, but it often serves the function of making a character seem more mysterious or eccentric.
In older (pre-1900 in North America; pre-1970 in the UK) fiction, a narrator may refer to a character (especially an older or more socially prominent character) by his or her surname. This is because at that time first names were much less commonly used socially than they are now; a young character may not even know the first name of an older character he or she is not related to. It was also common in that time to blank out the names of real people to avoid lawsuits and the like.
A common joke is to do The Un Reveal on the full name.
This trope can be somewhat justified, however, due to Nominal Importance: It's difficult enough for the writers to come up with good names for the main protagonists; it would be incredibly painstaking to come up with equally good names for a cast of characters who exist solely as part of the setting and serve no further purpose to the narrative.
Another common variation is for a series where a child is the main character to have parents only referred to as Mom and Dad.
Sometimes, a main protagonist will have no name to add to their mystique.
The link also provides more examples of anime characters who don't have names/surnames.