Jack turns out in the end to be Lady Bracknell's nephew, the son of her sister. Therefore, he and Gwendolen are first cousins. As first cousins were allowed to marry in England in the 1890s, that is not a problem. It was completely accepted among the upper classes and royalty (leading to too much in-breeding and, hence, introducing hemophilia into Europe's royal houses).
Wilde, though a member of the middle class (what most Americans would call the upper class), was a socialist, so he is satirizing valuing a person more for their bloodlines than their actual worth as a human being. Mrs. Bracknell is a snob who wants to protect Gwendolen from the "pollution" of marriage to a man of unknown family who was found abandoned as a baby at a London train station.
Jack is a worthy person, as evidenced by himself—the lively way he acts, his intelligence, and his good qualities—not his ancestry. Wilde skewers the people like Lady Bracknell who can not see this and trust bloodlines more than the evidence before their eyes. Lady Bracknell's rejection of Jack because of his unknown background is just as a flimsy and vacuous a reason to reject a person as Gwendolen's insistence that she can only marry a man named Ernest (which also happens to be Jack's real name). Wilde may also be having fun with the growing awareness by the 1890s that marrying a first cousin is probably worse, genetically speaking, than introducing some robust new lower-class blood into the Bracknell gene pool.
When Jack asks Gwendolen to marry him, he is unaware of any familial relationship between them. Jack's background is uncertain, as he knows nothing of his biological parents. He has been raised as an adopted son by Mr. Thomas Cardew, who found him as an infant in a handbag at the train station and christened him after the destination on his ticket at the time, "Worthing." Yet, it is precisely the difference in Jack's and Gwendolen's family connections that Lady Bracknell cites as a reason for them not to get married.
You can hardly imagine that I and Lord Bracknell would dream of allowing our only daughter—a girl brought up with the utmost care—to marry into a cloak-room, and form an alliance with a parcel? Good morning, Mr. Worthing!
In the 1800s, marrying someone of equal social stature was incredibly important. So, when Jack is revealed at the end of the play to be Algernon's brother, making him Gwendolen's cousin, Lady Bracknell is more accepting of their engagement, as he now has the same social class and background as her daughter. This was thought to be more "proper."
In fact, in the 1800s, and long before then, cousins marrying each other was an incredibly common practice. We see this time and again in literature of the time, such as in Pride and Prejudice , in which Mr. Collins asks Lizzy Bennet to marry him, even while calling her "cousin Elizabeth" in his proposal....
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Edgar Allan Poe even married his own first cousin when she was just thirteen.
Marrying your own cousin was thought to keep wealth within bloodlines and strengthen a family's stature and genetic purity. In addition, for anyone not living in an urban area, there simply weren't that many people around, and many of the people who lived in your town were likely part of your biological family.
Later in the 1800s, travel among rural areas became more easily accessible and there was more scientific evidence and awareness of the birth defects that occurred in the children of incestuous marriages. This resulted in a shift away from the practice of cousins marrying each other, and now, it is banned in many countries.
However, it's possible that Wilde intends us to be aware of the incestuous nature of Jack and Gwendolen's marriage. His play was written about twenty years after the practice of marrying cousins started being called into question, and as the entire play is a satire of the trivialities and manners of the upper classes, he may be highlighting the absurdity of Lady Bracknell's disapproval of Jack and Gwendolen's marriage when they are of different social statuses only to later approve it when she discovers they are cousins.
Further Reading
In Oscar Wilde's play The Importance of Being Earnest, Act III, it is discovered that Jack Worthing's father's name is, in fact, Ernest. This, in Wilde's own comical way, would then make Jack "worthy" of inheriting his father's name and, for that reason, he is now a "real" Ernest.
Algernon is also the son of Jack's father which, making for the ironic nature of the play, makes him also "legally bound" that Algernon's name will also be Ernest, if desired.
Lady Bracknell: I am afraid that the news I have to give you will not altogether please you. You are the son of my poor sister, Mrs. Moncrieff, and consequently Algernon’s elder brother.
Since Algernon is the cousin of Gwendolen and the brother of Jack, it is then true that Gwendolen and Jack are, indeed, cousins. However, this is not an issue in the 1800's. When we compare this subtopic to how it is treated by Margaret Mitchell in her novel Gone with the Wind, for instance, the characters of Melanie and Ashley become engaged and marry precisely because they are cousins and Ashley gives a lot of importance to marrying "within the bloodline". In this day and age that may sound ridiculous and even somewhat gross, but it was the idea back then.
Yet, bloodlines have nothing to do with Jack's decision to marry Gwendolen. They are quite unaware of their connection until the end of the play.