Objection 1. It seems that this name "God" is communicable. For whosoever shares in the thing signified by a name shares in the name itself. But this name "God" signifies the divine nature, which is communicable to others, according to the words, "He hath given us great [Vulgate: 'most great'] and precious promises, that by these we [Vulgate: 'ye'] may be made partakers of the divine nature" (2 Peter 1:4). Therefore this name "God" can be communicated to others.
Objection 2. Further, only proper names are not communicable. Now this name "God" is not a proper, but an appellative noun; which appears from the fact that it has a plural, according to the text, "I have said, You are gods" (Psalm 81:6). Therefore this name "God" is communicable.
Objection 3. Further, this name "God" comes from operation, as explained. But other names given to God from His operations or effects are communicable; as "good," "wise," and the like. Therefore this name "God" is communicable.
On the contrary, It is written: "They gave the incommunicable name to wood and stones" (Wisdom 14:21), in reference to the divine name. Therefore this name "God" is incommunicable.
I answer that, A name is communicable in two ways: properly, and by similitude. It is properly communicable in the sense that its whole signification can be given to many; by similitude it is communicable according to some part of the signification of the name. For instance this name "lion" is properly communicable to all things of the same nature as "lion"; by similitude it is communicable to those who participate in the nature of a lion, as for instance by courage, or strength, and those who thus participate are called lions metaphorically. To know, however, what names are properly communicable, we must consider that every form existing in the singular subject, by which it is individualized, is common to many either in reality, or in idea; as human nature is common to many in reality, and in idea; whereas the nature of the sun is not common to many in reality, but only in idea; for the nature of the sun can be understood as existing in many subjects; and the reason is because the mind understands the nature of every species by abstraction from the singular. Hence to be in one singular subject or in many is outside the idea of the nature of the species. So, given the idea of a species, it can be understood as existing in many. But the singular, from the fact that it is singular, is divided off from all others. Hence every name imposed to signify any singular thing is incommunicable both in reality and idea; for the plurality of this individual thing cannot be; nor can it be conceived in idea. Hence no name signifying any individual thing is properly communicable to many, but only by way of similitude; as for instance a person can be called "Achilles" metaphorically, forasmuch as he may possess something of the properties of Achilles, such as strength. On the other hand, forms which are individualized not by any "suppositum," but by and of themselves, as being subsisting forms, if understood as they are in themselves, could not be communicable either in reality or in idea; but only perhaps by way of similitude, as was said of individuals. Forasmuch as we are unable to understand simple self-subsisting forms as they really are, we understand them as compound things having forms in matter; therefore, as was said in the first article, we give them concrete names signifying a nature existing in some "suppositum." Hence, so far as concerns images, the same rules apply to names we impose to signify the nature of compound things as to names given to us to signify simple subsisting natures.
Since, then, this name "God" is given to signify the divine nature as stated above (Article 8), and since the divine nature cannot be multiplied as shown above (Question 11, Article 3), it follows that this name "God" is incommunicable in reality, but communicable in opinion; just in the same way as this name "sun" would be communicable according to the opinion of those who say there are many suns. Therefore, it is written: "You served them who by nature are not gods," (Galatians 4:8), and a gloss adds, "Gods not in nature, but in human opinion." Nevertheless this name "God" is communicable, not in its whole signification, but in some part of it by way of similitude; so that those are called gods who share in divinity by likeness, according to the text, "I have said, You are gods" (Psalm 81:6).
But if any name were given to signify God not as to His nature but as to His "suppositum," accordingly as He is considered as "this something," that name would be absolutely incommunicable; as, for instance, perhaps the Tetragrammaton among the Hebrew; and this is like giving a name to the sun as signifying this individual thing.
Reply to Objection 1. The divine nature is only communicable according to the participation of some similitude.
Reply to Objection 2. This name "God" is an appellative name, and not a proper name, for it signifies the divine nature in the possessor; although God Himself in reality is neither universal nor particular. For names do not follow upon the mode of being in things, but upon the mode of being as it is in our mind. And yet it is incommunicable according to the truth of the thing, as was said above concerning the name "sun."
Reply to Objection 3. These names "good," "wise," and the like, are imposed from the perfections proceeding from God to creatures; but they do not signify the divine nature, but rather signify the perfections themselves absolutely; and therefore they are in truth communicable to many. But this name "God" is given to God from His own proper operation, which we experience continually, to signify the divine nature.
Source: Summa Theologica (New Advent)