Salam
It maybe that the phrase is not a formality but forms part of the first verse and in case of Surah Fatiha is the first verse.
Keeping in mind this hypothesis, we will try to see the meaning of all the phrases of this in Quran.
Surah Fatiha: "It is through the name of God that all praise is for God". The name of God makes us know God.
Surah Baqara: "It is through the name of God that is Alif Lam Meem - that there is no doubt that that book - the Quran, is a guidance to guarding." This means with attaching ourselves to God's Name through Taqwa - the Quran becomes an undoubted guidance from God.
Surah Auli-Imran: "It is through the name of God that is Alif Lam Meem - that God who there is no God be he the Living - revealed the book as a confirmer to what is before it and revealed the Torah and Injeel" - this can be saying Mohammad (s), Musa (a) and Isa (a) who revealed these books - are given God's Name and are his name - that God revealed these books through these who are his name.
Surah Nisa: "Through the name of God - Oh humans, guard (regarding) your Lord who created you....", this can be saying that you should utilize God's Name to guard regarding your Lord and the continuation of the aya hints that morality is from God's Name and light and he establishes all rights through it including kinship and family relationship.
Surah Ma'eeda: "Through the name of God - O You who believe fulfill your obligations/bonds/agreements", this can be you should hold to God's Name and through it perform the obligations.
Surah Anaam "It is through God's Name that all praise is to God that created the heavens and earth and (it is through God's Name) that he placed the light and darkness....", it may saying addition to the fact all praise is to God through his name, that darkness and light take form through his name.
Surah Anfal "It is through God's Name they ask you about the spoils of war, say the spoils of war are for God and the Messenger...", this maybe saying that the Messenger is God's Name and they can ask you who is God's Name and the spoils of war surely are for God and his Messenger....
Surah Yonus "It is through God's Name that is Alif Lam Raa that these are signs of the wise book..." this can be saying that Quran designates God's Name and centering around it and through it, the wisdom of the book are proven.
Surah Hud "It is through God's Name that is Alif Lam Raa that the book becomes clear in it's signs and then explained from a wise aware...." , it may be saying that the Quran through the help of God's Name becomes a book of clear insights and signs and then is further explained through God's Name from God.
Surah Yusuf "It is through God's Name that is Alif Lam Raa that these (verses) are signs of the clear book...." It can be saying that God's Name makes the following verses signs of the clear book. This means Yusuf and Yacoub are examples of God's Name.
Surah Raad "It's through God's Name that is Alif Lam Meem Raa that these are signs of the book and (through God's Name) is the truth from your Lord..." This can be addition to Quranic signs saying the Quran itself through God's Name is the truth, and in a way saying Quran is God's Name too and that is the truth.
Surah Ibrahim "Through God's Name -that is Alif Lam Raa- God revealed the book to you so that you take people out of darkness to the light...", this can be saying the holy spirit is God's Name and Mohammad (s) understands the book through being God's Name and hence can bring people through it from darkness to the light having higher understanding of it and complementing through his Sunnah.
To be continued....