The way I see it, Mewberty is basically "magical puberty," a metamorphosis that signals Star's body is ready and able to channel its own magic ("dip down").
The "Butterfly form" that Mewberty induces is some kind of magic-enhancing transformation that mature Mewmans like Moon can use at will. Moon uses it when performing a big feat of magic in "Game of Flags," then again for the big fight in "Starcrushed." It's also notable that her wings have a diamond pattern, like her cheek emblems, which the book confirms are also related to the Butterfly family's magic. Star's wings lack a pattern, even when she was "Mewbescent," but that makes sense as her wings are still immature.
It seems likely that the raw animal lust that overwhelms Star during Mewberty is a one-time thing, a side-effect or symptom of the magic surging through and transforming her immature body. If Mewmans go into "heat," which is to say they have an estrus cycle rather than a menstrual cycle, Mewberty/the Butterfly transformation probably isn't related to it.
Mewmans still undergo a human-like puberty/development stage, and Mewberty seems to just be one (dramatic) part of it.
Star, like all future Queens before her, was given the Wand at 14. Continuing the assumption that Mewberty is basically "magical puberty," and that Mewberty is just part of Mewman puberty, it seems likely that Mewberty typically happens at least by age 14, so future Queens are given the Wand at around the same time that their bodies are ready to start handling magic. If her lack of physical development at 14 is any indication, Star could be a late bloomer in terms of her Mewberty too.