Who is Caroline Bingely talking about when she says "a family trait, I think"; is Caroline talking about Lizzy's family or Mr. Darcy's in Pride and...

Having searched the text to confirm my suspicion, I find
you have conflated two events and confused the quotations. Above, I've placed an
excerpted quotation from the contextual event you identify. In it, as you'll see,
Elizabeth actually says:

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"Mr. Darcy is not to be laughed at!" cried
Elizabeth. "That is an uncommon advantage, and uncommon I hope it will continue, for it
would be a great loss to me to have many such acquaintances. I dearly love a
laugh."

This occurs at
Netherfield after Jane has caught a cold (at her mother's insistence!) and while
Elizabeth is there to care for Jane. The entire party, including Jane, is gathered after
dinner in the drawing-room. Miss Bingley flirtatiously asks how to "punish" Darcy for a
"shocking" commentary on the ladies' motive for walking about the room. Elizabeth says
the way to punish a friend is to choose a foible and "Tease him--laugh at him." This
leads into the part of the conversation (Chapter 11) you cite as the context of the
quote attributed to Miss Bingley: "a family trait, I think," as you put it. Let's jump
ahead a bit and sort this out.

Deeper into the novel, in
Chapter 45, Elizabeth, along with her Aunt and Uncle Gradiner are sharing hospitality at
Pemberley where Darcy is eager to have Elizabeth and Miss Georgianna Darcy become
friends. The entire party, including Georgianna, is spending time with Darcy at
Pemberley, so Elizabeth is again in company with Miss Bingley. Darcy enters the room and
Miss Bingley alludes to the Militia regiment that Wickham was in while stationed in
Meryton. This allusion is meant to remind Darcy of the most egregious grievance he might
have against Elizabeth and, by extension, her family. Here, Miss Bingley
says:

"Pray,
Miss Eliza, are not the ____shire Militia removed from Meryton? They must be a great
loss to your family."

The
import of this statement is, as you now know, to degrade Darcy's opinion of Elizabeth.
The meaning of this snide remark suggests how poorly the other girls in the Bennet
family behaved in relation to the regiment (flirtation was socially frowned upon as
indecorous) and how fond Elizabeth had been (though no longer was) of Mr.
Wickham.

So now, let's sort your question out. (1) Your
quotations are imprecisely recalled (that's ok). (2) You join two very distant events
together in your memory. (3) The context of Miss Bingley's remark is Pemberley on
Elizabeth's trip with the Gardiners, not Netehrfield when Jane is ill. (4) Miss Bingley
is speaking of the Bennet girls. Now we can answer you specific question of whether Miss
Bingley means the Bennet family or the Darcy family. The answer is, technically,
neither. The quote is incorrect.

In a conversation in which
Darcy has no part, as he has been out of the room, Miss Bingley is speaking to Elizabeth
about the Militia and the Bennet family hoping that Darcy will hear, remember their
near-scandalous behavior and turn against Elizabeth because, of course, Miss Bingley
wants to marry Darcy. She is referring to the Bennets but in a wholly different context
and time.