To aid in explanation, an example might help. I'm going to pick on Dr. James White, not because I dislike him - just the opposite. He can be a little condescending at times - and I'm certainly not free to point fingers at others in this respect - but his books and debates were instrumental to my understanding of how to construct several good arguments for Calvinism and Protestantism. But I do have some difficulties with his view of the Trinity. I haven't read his book on it in a few years, but I think his "brief definition of the Trinity" in this article is a fair representation of it. He begins by stating:
The doctrine of the Trinity is simply that there is one eternal being of God - indivisible, infinite. This one being of God is shared by three co-equal, co-eternal persons, the Father, the Son, and the Spirit.
That is, the "one being of God" is simply a single set or list of attributes. Later in the article, he defines "being" as "what something is." God is "one what, three who's." The Father, Son, and Spirit "share" the attributes of eternality, goodness, etc.
The question is, what does Dr. White think "share" means? When he refers to "being" as "what something is," he continues to say "yet we also recognize individuals within a classification." Does he mean, then, that "being" refers to the class or genus under which some subset of "individuals" are members? That is, is "being" a universal which covers a set of particulars which "share" the same set of predicates or attributes? If so, then White believes in three Gods, for the Father, Son, and Spirit would be the "individuals" we would "recognize" as each individually possessing the attributes of eternality, goodness, etc. This interpretation of "share" would make sense in the context of the article, where Dr. White refers to the "being of man," rock, dog, cat, etc. in the same breath as the "being of God."
Perhaps he would contest the "three Gods" conclusion, but surely he would agree that there are many men, rocks, dogs, and cats - there are a plurality of these individuals. On his view, are there not likewise a plurality of Gods, then? If "God" refers to the divine nature, it would seem so. The Father is God (eternal, good, etc.), the Son is God (eternal, good, etc.), and the Spirit is God (eternal, good, etc.) just as Tom is human, Joe is human, and Billy is human. If this is what Dr. White means, though, would be imprecise for him to say God is "one what, three who's." Rather, God is one "what" - i.e. the definition of the genus or class or set or universal or attributes by which an individual is classified - but there are three Gods. Even on this view, God is not Triune; the Trinity are God[s]. Similarly, human is not "triune;" Tom, Joe, and Billy are human. God is not the subject and the persons the predicates. It's the other way around.
This view would be close to generic unity, the obvious difference being that Dr. White has asserted the definition proper of "God" is the attributes of the divine nature as opposed to the person of the Father. This begs an interesting question: Dr. White is a monotheist, presumably because of the reasons I outline above (God is "one what"). But given his references to the being of man et. al., is Dr. White a monoanthropist et. al.? He could, of course, consistently say "yes" and I would have to argue on other grounds why I think he should abandon his definition of God as a set of attributes. But I would like to hear whether this question gives him any pause.
Another possibility that would seem to be less likely is that Dr. White thinks the Father, Son, and Spirit "share" the attributes in a sense differently than I have outlined above; namely, that each person is not Himself eternal, good, etc. but are rather collectively so. If this is what he meant, his allusion to humans, rocks, etc. is very misleading, for as an individual human, I have my own rational and moral faculty, body, etc. I don't "share" it except in the sense that these attributes may be predicated univocally - though distinctly - of another human as well as myself. Thus, I don't "share" it except in the sense that I am a member of the same class or genus as Tom, Joe, and Billy. All of this leads me to believe Dr. White can't mean "share" in this sense, though even if he did, I don't know how he could avoid Sabellianism or some bizarre view that the persons are incomplete parts of God. If there is only one subject x of whom or which "eternality, goodness, etc." may be predicated, and the Father, Son, and Spirit are all said to share in these attributes, the conclusion that x = Father = Son = Spirit or x = Father (1/3) + Son (1/3) + Spirit (1/3) cannot be avoided.
This latter possibility is just what numeric unity is and entails. As soon as one affirms that the set of attributes (eternality, goodness, etc.) may be predicated of more than one subject - e.g. person - he admits that there are multiple members or individuals of the genus which is defined by that set of attributes. Each member himself or itself possesses the set of attributes. [How such are possessed is a separate question, so we need not get into eternal generation in this post.] But this is generic unity, not numeric unity. This is unity according to the members being classified under one genus. In this case, we say that the Father, Son, and Spirit are each individually "divine" (Dr. White would say "God"). Divinity is predicable of the Father Son, and Spirit, and they don't "share" divinity in any other sense. [They work together, obviously, but that too is a separate question.]
Simplified, then, the difference between numeric and generic unity is this: on numeric unity, there is only one subject of whom some set of attributes is predicable; on generic unity, there are multiple subjects of whom some set of attributes is predicable. Thus, if the set of attributes which Dr. White defines for "God" - for him, the definition proper - and I would define for "divinity" or "the divine nature" can be predicated of the Father, Son, and Spirit as distinct persons (subjects), then there numerically are three "Gods" (Dr. White) or "divines" or "divine natures" (me), not one.
I suspect most Trinitarians would be comfortable referring to the Father as divine, the Son as divine, and the Spirit as divine. But for some reason, they avoid the following conclusion or contradict themselves elsewhere: because there are three such subjects, it follows there are three such divine natures. Therefore, numeric unity is false. That is the only way the individuals can each be considered as a member of the class or genus of divinity without conflation or contradiction.