Any given name, considered as an abstraction, is a concept. Understanding it in a Fregean sense, a reference isn’t required to give the name – the concept - a meaning. You can think of Medusa and you understand the name has meaning despite being unable to point to the actual being you understand as Medusa. You can also have the same sense applied to different names because their reference is the same; that’s the case of Light Yagami and Kira. As anyone familiar with Death Note will notice, the sense attached to each of those names is completely different and so it’s their importance.
Light Yagami, as a concept, comprehends the exemplar student who wants to follow his father’s footsteps into law enforcement, who helps his younger sister, who dates remarkable girls, who excels at school. A good son, a good student, a good person. That is Light Yagami’s definitional structure, the conditions that need to be sufficed for one to be identified as Light. On the other hand, the necessary conditions for one to be identified as Kira are another set of requirements that does not consider Light’s definition in order to exist. The sense conveyed by Light Yagami and the sense conveyed by Kira can and are understood as different sense with the same referent.
Kira, as an abstraction, means a person with unknown supernatural powers and whose goals, motivations, methods and descriptions may vary. Kira is introduced as a vigilante killer but his definition isn’t restricted to it because the simpler concepts that characterize the abstraction of Kira are not specific enough; the set of characteristics applied to Kira could be those applied to any other name. If we have no concept, what can be said of its use? It’s from this vague definition that the power of being Kira comes from.
What is Kira? What makes one Kira? Could Kira have godly powers? That knowledge is restricted to very few people. There is nothing in the concept of Kira that contradicts most interpretations of to what the name Kira could be applied to.
Kira is allowed to become the God of the New World because there is nothing that disproves that equivalence. Not only that, the God of the New World in itself is a concept that requires a definition and that definition isn’t restricted to being Light Yagami. If its definition is that the God of the New World is the exact same being who behaves exactly like Kira does, to say Kira isn’t the God of the New World would be wrong. But Kira isn’t defined as being Light in the world of the series (we, as an audience, know but the characters are working with a broad concept that has little to no definition). The same issue arises from the equivalence Kira = Justice. If Kira is defined in terms of something or someone who punishes evildoers, one can argue that, in some sense, his actions are just without arguing Light Yagami himself is Justice.
What the detectives of the series do is to take away the vagueness of that definition in order to take away Kira’s power. The more they limit the definition of Kira (it’s a human, it’s a male, it’s a killer, it’s a Japanese person, it’s someone in the task force, it’s the second L), the more they limit the possible equivalences. If the God of the New World means Kira and Kira means a Japanese student who is nothing more than a vigilante killer, it’s quite unlikely there will be an equivalence with abstract ideas such as Justice because it would be to say Justice is a Japanese student who is a vigilante killer. All the Wammy’s detectives recognize the power of the usage of an undefined name and they not only humanize the idea of Kira and give it form but they strip him of his power with each characteristic added as a requirement to fulfil that concept.