From Britain or Brittany
A modern creative spelling of Brittany, which originally referred to people from the region of Brittany in France or the island of Great Britain. The name gained immense popularity in the 1980s and 1990s, leading to various phonetic spelling adaptations.
Less common today
Britanee isn't in the latest US Top 1000. It last peaked at #6,545 in 1991.
US births per 5-year period, 1990βpresent (SSA data).