13th Age is a game so different from D&D versions I've played (3-5, including Pathfinder) that I just don't know where to start with what I'm enthusiastic about, but the main thing I was thinking when writing this post was that the character has a lot of times between little and nothing to do with its class.
Something that used to set me back from class-based games was precisely the rigid class/archetype thing, with multi-classing being a half-solution attempt to more flexible systems such as Ars Magica, World of Darkness or Fading Suns, in which you get a set of points for skills, talents, etc and you make whatever you can with them. Of course they have archetypes too, but the archetypes are either character concepts or lore things, not only creation constraints.
In some cases, the archetype includes personality traits because they're educated in a particular way and culture (just like us). That's the case of werewolf tribes or Fading Suns noble houses.
In other cases, it's because they're chosen by a particular group, such as Vampire clans.
The point is, there's an easy, real-world related explanation for those traits.
In the D&D versions I played, classes defined not only what were the "right" skills of other people of your class (read, class-skills) but how many points you had, so rogue and bard classes had the more general life experience of all, while fighters were usually semi-ignorant, socially challenged brutes because they hadn't the points not to be.
In 13th Age all classes have the same points because thy're not skill points but background points. You have not Perception +3, but you spent half your life as explorer in the Imperial army, so you have a Explorer in the Imperial Army +4 background.
The point idea to backgrounds may even give meaning to what have you done with your life most of the time, and solves a couple problems: one, everybody have a past, being a soldier gives you a different experience than being a rock star, but not less, only different experiences. The second problem is the typical skills importance in RPGs. Some RPGs have so many combat/survival oriented traits that you need a lot of points to be somewhat good at them, while it's easier having a social/intellectual character. Usually that's no problem because most players (as far as I've met) are not the Geyperman type, but I remember that a friend, using the skill descriptions on World of Darkness books, had not enough points to describe what he could do - being a member of army's special forces.
Anyway, on with 13th Age's backgrounds. The point I want to make is that everybody have different experiences based on their past and profession, not that much or little. I know, in real life people do have different experience levels because some people learn faster than other, but I'm not saying that 13th Age's more realistic, but that is more flexible, character centered, than other systems.
Setting backgrounds aside, let's talk about the unique thing. Every player character must have something that sets them aside from the rest of the members of the same class and race. It can be something simple but when I've played, most players have more complicated ideas that are somewhat difficult to imagine in other systems, such as being a dragon trapped in human form or the only survivor of an old battle.
Finally, there's the icon relationships. They provide a connection with the great schemers in the world (or at least the setting) This is a bit more at fault, because not all the character types are going to be related to organizations. As for a system thing, it helps relate what motivates the characters with what happens in the adventures and with the world at large.
The great thing of those three traits is that they have nothing to do with your character's class. I wanted to use a couple examples to illustrate, so let's start with The Mountain's son. That character is a dwarf because it's in its story, as for the backgrounds, I'd say at least 4 points are in something like Dwarf Miner in the North Mountain Range and there's probably some in Dwarf Rock legends because he wants to know what means its unique. Of course, that is being born of a geode. What class is he? He could easily be a barbarian or a fighter as a lot of dwarves used to pickaxes, or a druid (possibly with terrain caster) to relate with the powers of the earth. He could even be a bard if he had time to travel enough, learning legends and stories.
A second example is the Crusader's Foundling. The kid that killed his demon-cultist family have that as unique, but what class is he? Paladin, fighter and barbarian all mix well with the violent twist, but he could also be a cleric (of the dark gods of punishment and revenge, of course, Inquisitor comes to mind) Changing the hammer detail, he could be a sorcerer related with the Great Gold Wyrm's power or a monk of the Crusader's Fist's Order, an order of warriors who only have in common the hate against demons, the will to use violence against them despite the odds, and a past without weapon or armor training. Also, making it a kitchen knife or even a sacrificial dagger, he could had develop his talents into being a rogue to infiltrate other cults or stealth missions to assassinate cult leaders while the main forces attack the cult's base. Or he could be a demon hunter, a ranger. In every case he knows about demons and cults and most probably have some skill at stealth (he freed the other children without the need to kill every cultist, after all)
Both characters' stories relate to what they know, what they do and what their allies are, but not to what class they are. I know, you can use the unique and the relationships in other games, but in 13th Age you have to, and that forces you to think about who your character is, how the world is and how they relate to each other, setting you in the mood.