I found online that kiki is the nickname for any name with a K sound. So names with K, or C’s that sound like k or Q. Like Angelique, or Dominique or Nicole or Katrina. Though Nikk, or Niki is more common for names like Angelique, Dominique, or Nicole.
Might have come from when they were children. Either Enrique or a younger sibling couldn’t properly say the name. They heard the “k” sound and pronounced the name “Kiki”. And it stuck.