I've found some interesting data about name Alizée itself.
Generally Alizée is quite new name in France (since 1980). It isn't also very popular name, there were/are 9731 Alizée in 1946-2006.
Lili had very big influence on popularizing her name,
in 2001: 1201 girls received name Alizée (64th most popular name in that year),
in 2000 it was just 561
Just look at
this and
this site
(they are in french of course)