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It is not the trinitarian language, it is the language with which Christ commanded his disciples to go and baptize:
Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: (Matt 28:19)
Trinitarianism interprets this to mean one God with three personages. The fulness of the gospel defines this as the Godhead:
God the Father, who is a physical being, the father of our spirits who is all powerful.
God the Son, who was Jehovah born in the flesh as Jesus Christ, our spirit brother, the eldest of all the spirit children of the father, to whom the father gave his power to create the earth, who suffered the atonement for the salvation of men.
God the Holy Ghost, who remains a spirit, and because he is a spirit he can communicate with our spirits and testify of the truthfulness of all things.
There are three Gods in the Godhead, but this does not make the LDS tri-theist, that would suggest that they worship three gods. Latter-Day Saints only worship one God, the God that Christ worshipped; God the Father.
President Gordon B. Hinckley gave this in an address to the church about the The Father, Son, and Holy Ghost in 1998:
Members of the Godhead Are Perfectly United
They are distinct beings, but they are one in purpose and effort. They are united as one in bringing to pass the grand, divine plan for the salvation and exaltation of the children of God.
In His great, moving prayer in the garden before His betrayal, Christ pleaded with His Father concerning the Apostles, whom He loved, saying:
“Neither pray I for these alone, but for them also which shall believe on me through their word;
“That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us” (John 17:20–21).
It is that perfect unity among the Father, the Son, and the Holy Ghost that binds these three into the oneness of the divine Godhead.
Upvote:8
While the Father, Son, and Holy Ghost are three separate personages, the name of the Father, Son, and Holy Ghost is the same. They are unified in purpose and will.
So what's in a name? Reputation. By extension, a name represents an agent or authority. Their reputation, authority, and agency (free will) is unified and is thus the same, even though they are separate beings.
Basically, saying "in the name of" is nearly synonymous with "by the authority of" or "with permission from."