Owen ranks #840 with 140 male registrations. The name is a Welsh masculine (from Owain, possibly meaning "young warrior" or related to the Latin Eugene's "well-born") that has been steady-to-rising on US baby registries for two decades.
The Welsh-Celtic register
Owen sits in the cluster of Welsh and Irish names that crossed into mainstream American use in the 2000s and 2010s: Finn, Liam, Aiden, Connor. The Welsh source carries specific cultural weight (the legendary Owain Mab Urien of Arthurian tradition) but most American owners pick the name on sound rather than etymology. On pet licenses, Owen lands disproportionately on shaggy-coated dogs whose appearance vaguely evokes the Welsh-mountain register: bearded collies, Tibetan terriers, and Welsh corgi mixes.
Sound and call-name fit
Two syllables, front-stressed (OH-en), with an open vowel opening and a soft N close. The name calls clearly outdoors and tolerates the affectionate household register. Owen also pulls a smaller cultural cluster from Owen Wilson (the actor) which lends a slight warmth-of-comedy register. See Welsh corgi names for the heritage cluster.
The counter-reading
The honest concern is the strong human-coding. The human Owen page shows top-50 SSA presence, and the dog will share call-name space with substantial numbers of human Owens. Households who want the Welsh-Celtic register with stronger pet identity might consider Oslo or Odin, both of which read more decisively as dog names.