The problem of clandestine (sometimes known as Fleet) marriages was resolved by the “Act for the better preventing of clandestine Marriages” more than two decades before Jane Austen was born. However, the results of its resolution can be seen in her novels, though it is possible that many of the characters (if they were living people) would have never given much thought to the reason behind why England required a betrothed couple to have a marriage license in order to be married.
So, the problem of clandestine marriages was that before the act I mentioned above, it was very, very easy to be secretly married in England as the law did not require a marriage license or that the wedding itself include witnesses or even an officiant. The result could become an administrative nightmare, particularly since these licenseless, officiantless, witnessless marriages were as legally valid as marriages that that had been performed with some combination of the above (as marriages typically were). This meant that a husband would have ownership of any property his wife had owned before their clandestine marriage, or that a person could marry one person in a very private ceremony, leave that person, and marry someone else who was unaware of the first marriage and who would presumably believe him or herself to be legally married, only they wouldn’t be. Or you could have someone claim that a marriage had taken place when it hadn’t. Of course my research also indicated that one of the motivating factors for this law was to prevent the heirs and heiresses of the wealthy from marrying without parental approval.
Getting back to Austen, it’s a good thing for Georgiana Darcy at least that this law was passed, otherwise her path to becoming Mrs. George Wickham would have been considerably shorter, and there would have been little or nothing to prevent Wickham and Mrs. Younge from claiming that she had married him, even if she hadn’t.