Changing your last name when you get married only seems to apply to Hyur and Elezen. All the other races have surnames that aren't changed when getting married for various reasons. Many of them have to do with keeping track of who the father/mother of the person is.
General Info about Surnames
Hyur: Midlander surnames are mostly based on professions, locations where people lived or some outward feature. Highlander surnames seem to be more "violent" and based on nicknames people earned in battle.
Elezen: The surnames are split up between the Wildwood and Duskwight clans. You will never find a Wildwood elezen with a Duskwight surname and vice-versa. All Surnames are also valid male forenames as well.
Miqo'te: Seeker surnames for males is pretty easy. They're either the head of a harem and are a Nunh or not and are a Tia. Females simply have their father's name without the clan letter. Keeper surnames work like Hyur and Elezen surnames work, but it's the mother's surname that is passed down instead of the father's. I could see males taking their wife's name for this clan.
Lalafell: Surprise! There technically is no surname for either Dunesfolk or Plainsfolk. Instead it's more of a second forename that follows each clan's particular rhyme scheme. So nothing gets passed down from the parents and nothing changes when Lalafell get married.
Roegadyn: Sea Wolf surnames consist of the father's surname with "syn" or "wyn" tacked on the end depending if they are male or female. The Hellsgard don't make use of surnames. The "surname" seen in-game is part of their forename.
Au Ra: Raen surnames come in two flavors. The Raen with "official" surnames are from the warrior and ruling classes. These are more then likely passed down the family line and are probably changed when people get married. The Raen with "unofficial" surnames is everyone else. Their names are usually something to do with their profession. It should be noted that Doman/Hingan Huyr and Roegadyn use the same naming patterns as the Raen do. Xaela surnames are their tribal names. So everyone in the same tribe has the same surname. The only reason I could think someone would change their tribal name if they got married was if they got married to someone outside the tribe and then went to join the other tribe. We do have a Sidequest that revolves around this.
The "primary sources" I got this stuff from. The Race Naming Conventions thread. The Lore Book has some stuff. And the FFXIV Name Generator Full List which includes how all the different names are "built" by the game.
Edit about Ishgard stuff: "de" is a title of nobility. All the high houses use this with their legitimate members. You'll also notice that all the Heavensward have this as part of their names, even though it's revealed in the lore book that not all of them are actually from noble houses. This was added to their names after they became members of the Heavensward.
The surname given to people who aren't born into a proper family in Ishgard is Myste (yes, like the NPCs from the DRK quests). Aymeric for all intents and purposes is a legitimate, trueborn member of his house, the house Borel. The previous Lord and Lady Borel had been unable to have children and were more then happy to adopt and treat Thorden's child as their own. Because everyone was "in one it" (and were happy to be at that!), the rumors that Aymeric was adopted didn't really go anywhere.