Women usually change their last name (their father’s first name) after they get married, attaching their husband’s first/last name instead. Is there any Islamic significance in this tradition?

Islam does not require a woman to change or amend her name when she gets married. This has basically to do with the traditions of a society. Today many societies follow this tradition. Arabs in the times of the Prophet (sws), on the other hand, did not follow this tradition. So when Ā’ishah (rta) got married to the Prophet (sws), her name did not change to Ā’ishah Muhammad (rta). It remained as it was before. Similar was the case with other wives of the Prophet (sws), and indeed with every Arab lady.