From what I've read Accomplishments formed an essential part of the education of a genteel young woman.
The purpose was not to bring pleasure or real improvement to herself, unless she was especially talented in say, painting. Accomplishments like music and drawing were embellishments to increase a young woman's attractiveness as wives for potential marriage partners and to make her an asset at social gatherings-singing or playing, etc.

(With ref. to M. Lane 'Jane Austen's World', p.48.)

Chapter 8 reads like a debate on The accomplishment woman; but I don't think it is just a delination of Bingley and Darcy but Caroline as well.
Caroline Bingley has a more rigourous idea of accomplishments than her brother : a thorough knowledge of music, singing, drawing and the modern languages, etc.
She probably believes she is describing herself for Mr Darcy's benefit. ;)
Yet Darcy also demands of the accomplished woman 'something more substantial, in the improvement of her mind by extensive reading' ch. 8.