I hyphenated my name-his name in order to show solidarity but still be findable, since I had published a number of scientific papers under my maiden name. He wanted to do the same (but in reverse, his name-my name) but the Navy couldn’t cope with the thought. The Navy doesn’t do hyphens, either, so it is always fun to watch corpsmen try to pronounce my name when they call me in for my medical appointments.
In Texas, while it is most common for women to exchange their last name for their husband’s, the cost of the marriage license includes a legal name change opportunity for all concerned. The man can change his, or the woman can change hers, both can change their names to something completely different as long as it is not obscene, or neither can change their names.
The only problem we have run into with names is that my son has two middle names, one from our family and one from his birth family. Social Security was flummoxed and could only permit all of his names if we hyphenated two of them so four names could fit into three slots.
And @HTDC, the tradition came about because marriage indeed was the transfer of ownership of a woman from her father to her husband for many centuries. This past 100 years has made changes to that legally in the US thanks in large part to the various mini-ERAs passed in the various states and such things as granting women the right to vote. However, society changes more slowly in some cases, and many countries women are still considered to be chattel, a form of personal property.