Home ruler
Enrigue is a rare variant of Enrique, which is the Spanish form of the Germanic name Henry. It is composed of the elements 'heim', meaning home, and 'ric', meaning power or ruler. While Enrique is widely used, this specific spelling is exceptionally uncommon.
Less common today
Enrigue isn't in the latest US Top 1000. It last peaked at #4,236 in 1990.
US births per 5-year period, 1990βpresent (SSA data).