Mike Baber wrote:No offense Mr. Somsel, but if I knew the type of genitive, I would not have asked the question.
Mike, we have been having a bit of fun at your expense, but the fact is that it really was, for all your bewilderment in raising the initial question, a "dumb question." We've been going back and forth in this forum and on its predecessor, the old B-Greek mailing list, on the matter of adnominal genitives and the tendency of some grammarians to divide and subdivide them into umpteen categories and subcategories, to the ultimate self-disparaging invention of the term, "Aporetic Genitive" -- meaning, "It doesn't fit into any of these gazillion subcategories of adnominal genitive we have 'discerned', and we still don't know what-the-hell sort of genitive it is" ("aporetic" means something like "stumped, at an impasse").
The categories and subcategories of the adnominal genitive are, by and large, devised in terms of how people think they should translate the construction. The only one of those categories that can really be discerned with any clarity is the genitive used with a verbal noun; there we distinguish between an "objective" genitive, e.g. ἡ τοῦ θεοῦ ἀγάπη, where we understand the ἀγάπη as directed toward God, and a "subjective" genitive, where we understand God as the one who loves.
But the fact is that every adnominal genitive -- every genitive noun that qualifies another noun -- is a structure without any distinct semantic value. The noun in the genitive case is linked to the other noun in the same way that one noun is linked together with another in English by a possessive-case form, by the preposition "of" or by compounding (with or without a hyphen). Thus ἡ τοῦ θεοῦ ἀγάπη can be Englished as "God's love" or as "love of God" or as "God-love." I think the reason why students get stumped by instances of an adnominal genitive encountered in their reading is that they can't see immediately what the linkage between the two nouns is. The fact, however, is that one can get no clues about what that linkage might be apart from the context of the construction in the text under consideration. There's nothing special or magic about the construction μορφὴ θεοῦ. One might play around with how we understand the noun μορφὴ here: is it used in an Aristotelian sense as 'the identifying formal quality that characterizes God and only God" or is it used in the sense of "outward appearance that may be deceptive"? But the phrase itself may be Englished as "God's form", as "form of God" or "God-form" -- or, if you prefer, with some word other than "form" for μορφή.
That is to say: there's nothing distinct about the phrase μορφὴ θεοῦ; its meaning is as obscure or clear as the corresponding English equivalents "form of God" or "God's form" or God-form.
Categorizing this geniive construction, determining "what kind" it is, is a waste of time. If you can't figure out what the meaning of μορφὴ θεοῦ from the Greek text and its larger context, grammatical lore is not going to help you. The more important question is probably, "What exactly does the writer mean by using the word μορφή?"
That may not be the kind of answer you're looking for, but I hope it will explain why you've gotten the kinds of answers you've gotten hitherto and why we've treated it as a "dumb question."