Rosemary is a female pet name at the intersection of the botanical trend, the vintage name revival, and the herb-as-name aesthetic. It fuses Rose and Mary into a single herb associated with memory, loyalty, and the Mediterranean garden. That's a lot of meaning for a pet to carry, but they manage.
Herb-Name Botanical Trend
Sage, Basil, Thyme, Rosemary: herb names have been crossing into both human and pet naming for years, driven by wellness and culinary culture. Italian Greyhounds and Whippets suit Rosemary particularly well; their lean, Mediterranean look matches the herb's origin landscape.
Vintage Appeal
Rosemary as a human name peaked in the 1950s and has been in slow revival mode since, following the same arc as Dorothy, Beatrice, and Mabel. See the human version at Rosemary for comparative context.
Counter-Reading: Rosemary's Baby Association
Rosemary's Baby (1968) is the name's most culturally charged modern reference. Most people won't go there immediately, but some will. If the Gothic overtone works for you, it adds interesting texture to an already layered name. If not, Rose is the simpler, warmer extraction with none of the horror overtones.