First we say אלקינו ואלוקי אבותינו, and the we break it down אלקי אברהם אלקי יצחק ואלקי יעקב . Why do we need both? Don't we know who the avos are?
I have read, but unfortunately forgot the source, that the specification of the avoth teaches us that the same G'd manifested itself in different ways to Abraham, Yitzchak and Yaakov. This teaches us that our relationship and how He manifests in our lives is unique, but these manifestations are not different gods, but the G'd of our ancestors, the One G'd.
This ties in to the answer of Yaacov Deane, who shows why the general (the One G'd of our ancestors), is followed by the specific (who was experienced differently by each of them).
There is a general principle that carries through into many areas of Torah called כלל ופרט (General or inclusive and particular or detailed).
This is alluded to in the names of the final parshiyot of sefer Shemot in Vayakhel and Pekudei. Vayakhel is the gathering of the whole congregation (ויקהל משה את-כל-עדת בני-ישראל), while Pekudei emphasizes the individual tribes, like in Shemot 38:21-23 which specify the tribes of Levi, Yehudah and Dan and like Rashi explains to 38:21 that the emphasis of Pekudei is enumerating the individual parts of the Mishkan.
The same principle of כלל ופרט is also enumerated in the beginning of Torat Kohanim (To parshat Vayikra) in the 13 Middot of the Torah from Rabbi Yishmael. It is the 4th quality.
Similarly, this idea is repeated in the opening of the davening which you cite. Avraham, Yitzchok and Yaacov are all called Av (אב), which can be understood as father or source. The G-d of our fathers/sources means it is coming from our perspective as individuals, but with recognition that we all have one source (שכולן מתאימות ואב אחד לכולנה ולכן נקראו כל ישראל אחים ממש), like is found in chapter 32 of Likkutei Amarim Tanya of the Alter Rebbe.
Essentially, it is the distinction between the wholistic and the individual. Or in other words, the idea of Tzibbur (the whole community) and individual.