Then why label those mortals in the first place? What is gained by doing this?
Because it can be a useful shorthand for the player
Because they're political affiliations and are relatively well defined. They're not really comparable to alignment at all. Alignments are not political factions, they're some bizarre mix of personality type and morality.
The various alignments
are relatively well defined, as I define them. I find thinking of them as political parties and/or moral compass descriptors quite fitting. Perhaps Meyers-Briggs personality types describes them to you better?
Gods follow cosmic principles because they are the emanations of such, and are bound by the principles of their creation and participation in the Nine Planes. This is wholly different than describing personalities of mortals. In this case I'm identifying cosmic forces that the Patrons and Powers identify with and draw power from. If I'm looking at mortal alignments as descriptors as Whigs or ETNP, the cosmic alignments are more like identifying phenomena that originate from the four forces (Gravity, EM, &c.)
This is confusing, using the same term for two different things. I did not explain that properly earlier, pardon me.
Why? I mean you can have your gods bizarrely fixed into nine stock personality types if you want, but that seems exceedingly limiting and hardly a sensible basis for the game in general. Why cannot the god of war be both impulsive and honourable?
Sure, Brisingr the Son of Summer can be described as such. Although he is bound to the Sublime Pattern rather than the Radiant Obelisk.
They have their own personalities based upon their characteristics and mythology. But, cosmically, I have five alignments rather than nine (those polar alignments I keep referencing). This allows for a wider net.
I mean if you want house rule things, "This spell cannot be used against the divine agents of the deity who granted it" does the same, without the weirdness of agents of two opposing gods who happen to share an alignment being immune to each other's powers.
Having the clerics of Bahamut - LG (Sentinel of the Horizon, The Shield of the Common) and Asmoder - LE (The Grand General, Lord of the Pit) with a major spell that doesn't affect the other side, in this case, is a feature not a bug.