The sense of “extraction” that we need here is:
extraction, n. 5.a. Of persons: Origin, lineage, descent.
Oxford English Dictionary.
So the “contrast” that Miss Wade refers to is between the descent of “the wife of your dear friend Mr Gowan” (that is, Minnie Meagles, nicknamed “Pet”, though at this point in the novel they are engaged and not yet married) and the descent of “this girl” (that is, Tattycoram) and herself. Pet knows her own lineage, but Tattycoram and Miss Wade do not know theirs. Pet is the daughter of Mr Meagles, a gentleman (that is, a man wealthy enough not to be obliged to earn his own living), but Tattycoram was a foundling (a child abandoned as a baby by her parents) brought up at the “Foundling Hospital in London”, and Miss Wade an orphan whose parentage was concealed from her. Tattycoram does not know her own family name, and even the name she was given by the Foundling Hospital has been erased by her employer:
‘By George!’ said Mr Meagles, ‘I was forgetting the name itself. Why, she was called in the Institution, Harriet Beadle—an arbitrary name, of course. Now, Harriet we changed into Hattey, and then into Tatty, because, as practical people, we thought even a playful name might be a new thing to her, and might have a softening and affectionate kind of effect, don’t you see? As to Beadle, that I needn’t say was wholly out of the question. If there is anything that is not to be tolerated on any terms, anything that is a type of Jack-in-office insolence and absurdity, anything that represents in coats, waistcoats, and big sticks our English holding on by nonsense after every one has found it out, it is a beadle. […] The name of Beadle being out of the question, and the originator of the Institution for these poor foundlings having been a blessed creature of the name of Coram, we gave that name to Pet’s little maid. At one time she was Tatty, and at one time she was Coram, until we got into a way of mixing the two names together, and now she is always Tattycoram.’
Charles Dickens (1857). Little Dorrit, book 1, chapter 2. Project Gutenberg.
This is why Miss Wade says of Tattycoram, “She has no name”.
Miss Wade’s parting shot to Arthur Clennam is in the form of conventional wish of good fortune for Pet, but from her “cruel face” we deduce that there is a sarcastic barb in it. In fact there are two. First, Clennam and Gowan had been rivals for the affection of Pet, so by alluding to the forthcoming marriage of his “dear friend Mr Gowan” she rubs Clennam’s nose in his romantic defeat. Second, Miss Wade herself had been jilted by Gowan, and she knows his character to be selfish and cruel, so that it is not “high good fortune” that awaits Pet but rather suffering at the hands of her husband.
‘I hate him [Mr Gowan],’ she [Miss Wade] returned. ‘Worse than his wife, because I was once dupe enough, and false enough to myself, almost to love him.’
Dickens, book 2, chapter 20.