Am I correct that 'cause' refers to cause of action?
But 'style' and its earlier meanings appear unfit?
Style in the sense of an official title or designation as in"‘the partnership traded under the style of Storr and Mortimer" (noun sense 1.4 here) is related to and probably derives from verb sense 2 at the same link:
Designate with a particular name, description, or title.
"the official is styled principal and vice chancellor of the university"
The origin of the word is as follows:
Middle English (denoting a stylus, also a literary composition, an official title, or a characteristic manner of literary expression): from Old French stile, from Latin stilus. The verb dates (first in style (sense 2 of the verb)) from the early 16th century.
My suspicion is that the Latin/French word for a writing instrument ends up being used for the act of using a writing instrument to place a name upon something, which in turn comes to mean the name written as a result of this act, which in turn comes to mean any name. Once this chain of multiple meanings is in place, the word migrates from Middle French to Middle English.
The preference for words of Latin origin, often via Norman Middle French, is typical of common law legal language because the English common law was established by the Normans whose elites administered the legal system and naturally favored words familiar to them rather than Germanic words favored by the common people.