Flower names are the oldest naming habit there is, and the easiest to fake. A name is tagged as a flower name here only when it names a plant someone can point at — the tag comes from a lexicon that records which plant, so it can be checked.
Why the list is shorter than you expect
Because sounding floral is not the same as being a flower. Names that merely feel soft and green are not in the list; if you want those, filter by sound instead of by style, and the catalogue will give you names classified by their letters rather than by an association nobody can verify.
Every count on this site is a tally of real registrations: pet licences and shelter intakes published as open data by New York, Seattle, Austin, Montgomery County, Cambridge, Zurich, Basel and three Australian councils. Popularity here is the average share a name holds across those registers, so a name common in a small city is not buried by one common in a large one. Nine of the eleven sources are North American, so read the ordering as naming in those places rather than worldwide.