The question is, does Queen Elizabeth (and sons) actually have a
"last" name?  A surname, in the customary sense of the word? 
Historically, monarchs had - and only needed- one name, but they had a
dynasty name, known as their "House". Thus, Catherine of Aragon, Anne
of Cleves, and so on.  English monarchs usually had numbers only,
though you do read of Richard Plantagenet (Richard the first, called
Lionheart) and a few others. The argument here is that Windsor is the
current Royal Family's "last name".  The opposition says they have no
actual surname, and that Windsor, the name of their Dynasty, is merely
being used by such of the Royals who choose to go into business where
the lack of a last name is awkward.  I read recently that Prince Harry
wants to use Wales at
school.  If he had a last name, why would he have to "use" anything
else?

You could say Mountbatten-Windsor is their last name or surname, in
which case I can understand a young man wanting something shorter and
snappier. But the official line is that you don't use a surname if you
have a "titular dignity" like Prince. The authoritative Burke's
Peerage reports the Queen's announcement  on "8 Feb 1960 that Her
descendants other than those enjoying the style, title or attribute of
HRH and the titular dignity of Prince or Princess and female
descendants who marry and their descendants shall bear the name
Mountbatten-Windsor"

http://www.burkes-peerage.net/Sites/Peerage/SitePages/page62-6a.asp

See also:

http://www.hist.umn.edu/Courses/Hist3151/peers/peers.htm