I am going to go on record and say that Tempering is not exclusive to Primals, and that the player characters (as well as a few others) are tempered by Hydaelyn.
While it is true that the word 'Temper' is only used in reference to what the Primals do, I believe this to just be a cultural thing used by the non-beast tribe people of Eorzea to designate a person claimed by a god. Hence why the Amalj'aa priest says that the player is claimed by another. While Ifrit says that they do not have the taint of another upon them, I believe that this is reference to other Primals like him.
Yes, I say "god" and not "Primal." "Primal" is another thing that is cultural. The Primals are the gods of the beast tribes, and the beast tribes do not call them Primals. This is further evidenced in the main story line during the Little Ala Mihgo portion where the Ascians showed how, and convinced the children to summon Ralhgr(spelling?) in the same manner as the Primals, which is the same as with the Moogle King. While you stop them from summoning their god, the leader makes a comment about how if he was summoned he would temper his people like the Primals do and how that was not the way to do things.
From this, I have taken that all beings that are not given actual life by the Life Stream in the game are, in fact, beings of Aether, and that all beings of Aether can cause the effect of Temperence upon those given life by the Life Stream. This explains why even minor dragons can do so, as the story leads to Dragons as beings of Aether.
All beings under the effect of a temperence exhibit the same universal symptoms.
1) Obedience.
2) Worship.
After this, the individual Aether being that marks the person alters them in whatever way they do. The player character is no different, as from the beginning of the game the were ordered by Hydaelyn to stand against Lahabrea and "The Darkness." While the player character does not speak, they do agree with Minfilia when Minfilia offers worship to Hydaelyn at the times she (or others) do. While the influence of Hydaelyn may seem more relaxed than the Primals, I believe this is due to the more overall goal of going against Zodiark and the necessity for patience. It is shown in the story line, as well, that even the tempered of the Primals can work with those not so to achieve their goals. It makes sense that the players doing what they do for others is not out of the ordinary in this case. As well, having a position of influence amongst people who can fight is more advantageous to battling an enemy like the Ascians than if the adventurer was just some dude that killed monsters.
It is said in various places in the game that champions of the Aether beings are often given special gifts by their gods. The Echo, I believe, is just Hydaelyns gift to her champion/s. I'm also pretty certain that already tempered people cannot be tempered by another. Someone (I forget who, pretty certain it was Ifrit) makes a comment about how the mortal form can't hold the power of more than one, which is why it doesn't work, not that they are protected from it.
In short, all gods are Aether beings, Primals are gods, Aether beings can cause the effect of Temper in living beings, and the player characters (and some major NPCs) are Tempered by Hydaelyn.
