Rabbi Y.B. Soloveitchik takes the possessive in “Elokei Avraham, Elokei Yitzchaq, vEilokei Yaaqov — the G-d of Avraham, the G-d of Yitzchaq, and the G-d of Yaaqov” in a sort of mystical way: that one can somehow take possession of the Creator.

I want to say something similar, but with a more rationalist presentation:

The posessive could be used to show that two things exist in relation to each-other in a number of different ways.