Is this something that predates back many civilizations ago? Or is this a relatively newfound trend?
In general, it is a relatively new trend of the last few centuries.
Keep in mind that surnames in most cultures are a relatively new trend. There was no name to drop upon marriage if you didn't have a surname. The Nordic cultures, for instance, historically used patronymic "surnames". These generally would not change. Emma Jonsdóttir does not cease to be Emma, Daughter of Jon upon marrying Erik Eriksson.
Similar practices existed among other peoples such as the Welsh, for example, well into the Early Modern Era. On the other hand, some cultures like Greek would have referred to the married woman as "wife of Erik".
In some other cultures with longer histories of surnames, women did not historically drop their surnames at all. The Japanese, for instance, typically retained their clan name; whereas some Chinese historically added their husbands' to their own. This isn't isolated to Asia however; up until the the Early Modern Era, Scottish women of the lowlands did not customarily drop their maiden names. Further back, Ancient Roman women did not change their nomen upon marriage, either.
The practice of dropping the maiden name in the English speaking world was established after surnames became common among the English people. Since that's sometime in the 13th and 14th centuries. So no, it did not predate civilisation.
Why is it that the maiden name is traditionally dropped when a woman is getting married?
This is a lot more murky. In the English tradition, it is often said to be rooted in women being quasi-properties of their husbands without a separate legal existence; and that therefore they take their husbands name to mark themselves as extensions of the man of the family. It is difficult to determine the veracity of this claim.
Generally speaking, however, I'm more inclined to argue that the practice of dropping maiden names occur in two situations:
- In societies that did not have strong, blood-oriented views of family. So upon marriage, women are seen to have joined a different family, and the husbands' names are adopted in recognition of their new family. In contrast, culture that valued blood lines (e.g. Japanese clans) held on to their own clan names.
- Societies that did not have surnames until relatively recent, which overlaps with point 1. Cultures with strong views of family tend to adopt a collective representative name. In those without, surnames tend to come about for identifying otherwise similarly named individuals. It would have been convenient, and would indeed make sense, to identify a wife by her husband's name ("Agnes who married the smith; not Agnes who married the carpenter"). Then as surnames became more established, then dropping maiden names became a traditional custom.