Kaius sits next to 4 phonetic neighbors — Caius, Kai, and Darius are the closest matches by sound. Most share Kaius's Latin-origin cluster, so the rhythm and vowel shape carry across the list.
You searched for names like
"rejoice, be glad"
"God-like"
"Basque masculine name, nature-connected"
"stream, small brook"
"free man"
"God is gracious"
"all wise (from Norse Alviss)"
"God has heard (via Samuel)"
"victory of the people"
"wolf counsel"
"divine reward (alternate interpretation)"
"elf warrior"
"love, romance"
"industrious, hardworking"
"motion, movement"
Similar sound: These names share phonetic patterns with Kaius — similar vowel sounds, consonant clusters, or rhythm.
Shared roots: Several names on this list have Latin origins, giving them the same cultural heritage as Kaius.
Rhythm: Kaius is a 3-syllable name. Names with the same syllable count tend to feel interchangeable when you're narrowing your list.