Sailor Moon (and friends) have a recognisable transformation style. Other anime—like Dragon Ball and Puzzle & Dragons X—parody or (likely) take inspiration from such transformations.
Is there a name for this style of transformation? It’s so recognisable that it feels like it should, yet I’m not finding anything.
Bonus: Did this style of transformation originate with Sailor Moon, or did it just popularise it despite there being prior art?
Yes, there is.
This is called a henshin shiin (変身シーン), which means "transformation scene."
The magical devices that characters use to accomplish them, such as Sailor Moon's brooches or the other sailor senshi's pens or sticks, are all called henshin aitemu (変身アイテム), meaning "transformation items."
A few related terms are henshin poozu (変身ポーズ), meaning "transformation pose" - that is, the final pose the character strikes when the transformation sequence is finished - and henshin no BGM (返信のBGM) or henshin kyoku (変身曲), meaning "transformation background music" or "transformation song" (although each of these musical pieces have their own title, such as『ムーン・プリズム・パワー・メイクアップ！』(Moon Prism Power, Make Up!), 『ウラヌス、そして、ネプチューン』(Uranus, and also, Neptune), or 『スターパワーメイクアップ！』(Star Power, Make Up!).
This style of transformation scene existed in the long-standing mahou shoujo (魔法少女: magical girl) genre that falls within the broader shoujo (targeted at the demographic of young girls) genre long before Bishoujo Senshi Sailor Moon came into existence in the 1990s. Magical girl manga dating from before Sailor Moon include panels representing it as well, but the primary reason for these extended scenes in traditional cel animation was to use up a good amount of time in each episode of higher quality, reusable cels that could reduce the amount of other cels needing to be hand-drawn to come out to about a 22-minute-long run time. Arrival scenes (where characters recite a trademark phrase) and attack scenes serve the same purpose. After the switch to computer animation, this time-saving repetition became less necessary from an animation company's production standpoint but, because such scenes had become iconic over the past decades' worth of magical girl series, they are still used in fully-computer animated series, such as Pretty Cure - and they still continue to serve the same purpose (to reduce the workload of the animators per episode), albeit if less crucial. Ironically, in many cases, these more recent scenes have stretched out considerably longer than they had ever been in the classic magical girl series (perhaps to poke fun at the genre, perhaps to show off what computer animation can do... perhaps both), even as the practical need for them has largely decreased.
What Sailor Moon did innovate was combining that decades-old genre with the sentai (戦隊: team-fighting) genre - after Sailor Moon, many other series copied this, leading many people without knowledge of the classic magical girls to assume that the genre is usually comprised of teams of superhero girls, but that was not the original style, in which either a) a single girl from another magical world came to this world, or b) a single ordinary girl in our world suddenly received magical powers (in most cases, to use for nichijo [日常: daily life] situations rather than for saving the world from monster or alien villains). Nor have all magical girls since Sailor Moon employed teams and extended transformation sequences.
For more details, see "Have the creators of Tokyo Mew Mew or Sailor Moon commented on the similarities?", "Why is “Puella Magi” equivalent to “Mahou Shoujo”?" and "Why do the senshi get a manicure before they get gloves?"
