Jane Austen carefully establishes equality between the two characters but not always similarity. Their positive qualities largely complement each other.

One quality they share in equal amounts is (unsurprisingly) pride. In Lizzie's case, she has a strong ego and is defensive about her family's position and reputation, not just their limited means. Lizzie would like to indulge her own interests in books as well as socializing, and is proud of her own learning and intelligence. This aspect of her pride sometimes causes her to seem cruel or at least thoughtless, as for example concerning Charlotte's marriage.

Darcy's pride stems from his social position, as he tends to place credence in rank. More than that, however, he has confidence in his own qualities as a leader and protector. While this confidence seems over-inflated, we later learn how he won it through his efforts to care for his little sister, including rescuing her from Wickham.

Another important area of equality is intelligence. Their initial encounter features clever banter, showing their verbal abilities are well matched. It takes longer for them to realize that each other's brain power goes deeper and that both are similarly serious people, as Lizzie in particular enjoys joking around. The ways each person grows are different, but Austen shows each as maturing in a way consistent with their nature, as well as growing toward each other, thereby helping the reader believe they will remain well-matched partners throughout their marriage.

Good question.  Here are a couple of equalities between Elizabeth and Darcy.

1. They seem to be equal in intelligence and wit. Elizabeth is an "accomplished" young lady, yet she does not brag about any of her talents. Darcy has a formal education, but he knows that Elizabeth can hold her own in a conversation with him or others of high intelligence.

2. They are equal in maturity. Darcy, from a young age, has had to care for his younger sister and a vast estate; so his maturity in comparison to Bingley's is much more advanced. Elizabeth, though she is not the oldest in her family, is certainly the most mature (many would argue that she is more mature than both her parents) and feels the weight of having to worry about her younger sisters' welfare and places in society.