Symbolism, Imagery, Allegory
Oliver’s resemblance to Agnes’s portrait at Mr. Brownlow’s house is what first gives Mr. Brownlow a clue that Oliver might actually be the son of Agnes and Edward Leeford. Of course (thanks for the suspense, Dickens) that clue isn’t shared with us, the readers, until later.
This is important: all we see is that Oliver feels some kind of connection to the lady in the portrait, and we’re left guessing what the connection actually is. In any case, it seems that the portrait represents the kind of gut connection people are supposed to feel for their families.
That kind of familial tie is especially important for Oliver, who never knew his parents. Even though he’d never seen his mother (at least, not that he’d remember), he still has some kind of instinctive feeling of attachment to her face in the portrait. But for characters like Monks, that sense of connection with family is completely broken, which makes Oliver’s heightened sense of connection with his mother’s portrait, and with Rose Maylie (who turns out to be his aunt) all the more important.