In short, it deserves a name, because it is important enough to devote a full book to this inequality: The Cauchy-Schwarz Master Class. An Introduction to the Art of Mathematical Inequalities, 2004, J. M. Steele.
First, and historically, the inequality progressively emerged in three bodies of works, one involving finite sums, the others with integral formulas, in one and two dimensions, where the notion of cosine might be less evident (back then).
On page 10 of this book, a glimpse of the story:
Augustin-Louis Cauchy (1789–1857) published his famous inequality in
1821 in the second of two notes on the theory of inequalities that
formed the final part of his book Cours d’Analyse Algébrique
This bound [in the form of integrals] first appeared in print in a Mémoire by Victor Yacovlevich
Bunyakovsky which was published by the Imperial Academy of Sciences of
St. Petersburg in 1859.
In particular, it does not seem to have been known in Göttingen in
1885 when Hermann Amandus Schwarz (1843–1921) was engaged in his
fundamental work on the theory of minimal surfaces [with a] need for a
two-dimensional integral analog of Cauchy’s inequality.
Often, objects are named afterward, as a recognition of the previous works.
I have discovered the book recently, and I believe it deserves attention, because of the many implications of this inequality, interesting tricks and subtle reasoning. For instance, the book offers an inductive proof in finite dimensions, which he deems novel. There are a few books on "inequalities", not so many on only one of them, especially when considered basic. Because this inequality is paradigmatic. The text:
is designed to coach readers toward mastery of the most fundamental
Cauchy-Bunyakovsky-Schwarz is used in a systematic way to open to the geometry of squares, convexity, the power means ladder, majorization, Schur
convexity, exponential sums, and the inequalities of Hölder, Hilbert, and