As I tried to indicate, I think the main issue is identification and ownership, which is rather "personal." And see below.
But to your question,
Jn.14:13-14 "Whatever you ask in my name, this I will do, that the Father may be glorified in the Son. If you ask me anything in my name, I will do it." cf. 15:16; 16:23.
We are coming to the Father, and asking for things as if we are the Son. Of course, none of us is the Son; but he has given us permission to use our identification with him, to presume on his rights/authority and access.
What does it mean to call on the name of the Lord? e.g Ps.116:13,17. Sure, it refers to a specific Divinity, and excludes others. There's something to be said for reading the baptismal formula in that way. But it seems to me that the matter of authority takes that "identifying" the one true God a step further. The name is applied to the person, but that can't be for identification AS the deity--something different is being stated. Something akin to these statements by Isaiah:
Is.65:1 "I was ready to be sought by those who did not ask for me; I was ready to be found by those who did not seek me. I said, "Here am I, here am I," to a nation that was not called by my name.
Clearly, there is a nation, a people, who were called by the name of God. Israel? Yes, indubitably, but not because they went planted in the world identified as "Jehovah" or any such. But as 2Chr.7:14 states, "my people who are called by my name." Now, the Gentiles will also be called by that name.
Is.43:6-7 "I will say to the north, Give up, and to the south, Do not withhold; bring my sons from afar and my daughters from the end of the earth, everyone who is called by my name, whom I created for my glory, whom I formed and made."
Again, even if one is indisposed to recognize that this is a promise that extends beyond the ethnic Jews, it is a text that recognizes that the people of God are recognized as those who
bear his name, in some sense.
Is.48:2 "For they call themselves after the holy city, and stay themselves on the God of Israel; the LORD of hosts is his name."
Here, the people (who are actually faithless) identify themselves by the city of God, by the body of the faithful, the residence of God, the person of God, the name of God. Is this not a "taking of the name of the Lord in vain?" Does this not presume upon a relationship that has been assailed by faithlessness and wickedness (cf.v22)?
A good bit of the imagery in the following chapters borrows from marriage. In marriage, the bride (traditionally) takes her husband's name. She is called by him to whom she now belongs, but not so as to be confused with his person. But this union brings about shared authority (among other things), which she borrows from him.