Why is it that the maiden name is traditionally dropped when a woman is getting married? Is this something that predates back many civilizations ago? Or is this a relatively newfound trend?
Inheritable family names may be considered a relatively new trend, only dating back to the dawn of the Renaissance in Europe, that is, their use on a large scale (say, 1500s). In some parts of Europe, they only were forced on the populace in the 1800s, or the 1920s, and Iceland still doesn't use them. On the other hand, in Asia they date back for thousands of years in some cases, like China, and still aren't used yet, as in Myanmar.
This is entirely a cultural trend, and each little culture group works it out for themselves. It isn't religious, or it would be largely the same throughout Christendom, and it wasn't. The Spanish notably didn't.
The Italians in the 1500's didn't. Negri's lists of participants in Nuove Inventioni di Balli lets us see that single women are known by personal name or two and family name (Antonia Viale) but married women by that plus "and" and the husband's family name in feminine form: Anna Sfondrata & Visconte is one of the Sfondrati who married a Visconte, while Anna Visconte & Arconata is a Visconte who married one of the Arconati. Lucia Visconte & Visconte was a Visconte who married a cousin.
So changing the maiden name to the husband's family name is not only relatively recent (depending on getting family names) but a minority habit. It's a cultural choice. Some, like the Italians or the Chinese append Mrs. Hisname like a title. Others, like the English and the Japanese, change the name of the woman.
Unless, of course, they change the name of the man.
In Japan, it was pretty normal that a man with just daughters would adopt her husband so the children would continue his family line, even if, technically, this made the happy couple brother and sister. They knew what was going on, and ignored that level of technicality.
In other cases, in England husbands would be convinced, usually by financial incentive, to append the wife's father's name to theirs, which is how you got hyphenated names. Not that they always used hyphens. Lord Byron was born George Gordon Byron, 6th Baron Byron, but later became George Gordon Noel, 6th Baron Byron, as taking the name Noel was the requirement to inherit a bunch of money from his wife's mother, whose maiden name it was.
LSS: It's only used by some cultures, especially the Anglophone, not by most. It is a Modern habit in Europe, and is disappearing on the legal front, though the whole Mrs. business is often retained socially.
SOURCE: Ingraham, People's Names, McFarland, 1997