Families consist of different members, who can have completely different financial histories.

Caroline, Louisa and Charles' money was earned by trade. The respectable family in the north of England is more deeply impressed on Caroline and Louia's minds, which imply to me that the family must have been something worth having for two ladies who were in the habit ... of associating with people of rank.

My guess is that Caroline, Lousia and Charles' father was the younger son of a family of gentry who earned a fortune through trade. In Regency England primogeniture was often practised, so if the Bingley family had an estate it would likely be left to the first-born son (with dowries for any daughters), while younger sons were expected to earn their own way. One of JA's brothers went into banking so this was an option open to the younger sons of the gentry. Sadly Henry Austen went bankrupt, but that doesn't mean others couldn't be successful. Another possibility was that the senior Mr Bingley married the daughter of a merchant who later inherited a large fortune. Caroline and Lousia just prefer to remember the family of gentry and ignore the new money.