Rhys sits next to 3 phonetic neighbors — Reese, Reece, and Rees are the closest matches by sound. Most share Rhys's Welsh cluster, so the rhythm and vowel shape carry across the list.
You searched for names like
"enthusiasm, passion"
"noble, illustrious"
"son of Owen"
"crossing, toll bridge operator"
"divine inspiration, fury, frenzy"
"broad clearing in the woods"
"hunter, one who hunts"
"shelter, dwelling"
"unity; the best; one"
"sorrowful (folk etymology via French)"
"high-spirited person"
"mythical bird of rebirth"
"peaceful, gentle"
"benevolent, kind"
"adornment, goodness"
"passion, fire"
"servant, messenger"
Similar sound: These names share phonetic patterns with Rhys — similar vowel sounds, consonant clusters, or rhythm.
Shared roots: Several names on this list have Old Welsh origins, giving them the same cultural heritage as Rhys.
Rhythm: Rhys is a 1-syllable name. Names with the same syllable count tend to feel interchangeable when you're narrowing your list.