Primals don't necessarily need to adhere to certain elements. That they are is merely coincidence, or is just the fact that they have traditionally been element. They might be able to use some aspects of some elements, but I don't really think they have to adhere to strictly to the rule of "They have to have an element."

A few corrections:

Alexander and Thordan probably have an affinity to Holy: Alexander because of course he does, and Thordan because a lot of his attacks all have to do with light or are rife with holy symbols.
Shinryu appears to be all of the elements. Sort of an amalgation of the powers of all the previous primals. It's unclear.

Lakshmi, Bahamut, Sophia, and Good King Moggle Mog, praise be unto his magnificent pom-pom, are all non-elemental, if only because they embody or exist as some sort of vessel of a certain thing. Sophia is Balance (though uses Thunder magic), Bahamut is Power (as well as being a god of dragons), Lakshmi is Bliss and a sort of Nirvana, and Mog is the embodiment of all things glorious, much like his soft, perfect fun, praise be unto him. Heck, even Ravana is less fire and more battle and destruction.