Here’s the start of the story ‘Sneewittchen’ as given by the brothers Grimm:
Es war einmal mitten im Winter und die Schneeflocken fielen wie Federn vom Himmel, da saß eine schöne Königin an einem Fenster, das hatte einen Rahmen von schwarzem Edenholz, und nähte. Und wie sie so nähte und nach dem Schnee aufblickte, stach sie sich mit der Nadel in den Finger, und es fielen drei Tropfen Blut in den Schnee. Und weil das Rothe in dem Weißen so schön aussah, so dachte sie: hätt ich doch ein Kind so weiß wie Schnee, so roth wie Blut und so schwarz wie dieser Rahmen. Und bald darauf bekam sie ein Töchterlein, so weiß wie Schnee, so roth wie Blut, und so schwarz wie Ebenholz, und darum ward es das Sneewittchen genannt.
Once upon a time, in the middle of winter, when feathery snowflakes fell from the sky, a beautiful queen sat and sewed at a window, which had a frame of black ebony. And as she sewed and looked up at the snow, she pricked her finger with the needle and three drops of blood fell onto the snow. And because the red looked so beautiful against the white, she thought: had I but a child as white as snow, as red as blood, and as black as this frame! Soon afterwards she had a daughter, as white as snow, as red as blood, and as black as ebony, and so she was called Little-snow-white.
Jacob and Wilhelm Grimm (1812). Kinder- und Haus-märchen, p. 238. Berlin: Realschulbuchhandlung.
This is the oldest printed version of the story, and here, it is clear that although the queen’s daughter has all three attributes from her mother’s wish, her name is just “Sneewittchen”.
In the Disney animated film (1937) there is a scene with the following dialogue:
Queen: Magic Mirror on the wall, who is the fairest one of all?
Magic Mirror: Famed is thy beauty, Majesty. But hold, a lovely maid I see. Rags cannot hide her gentle grace. Alas, she is more fair than thee.
Queen: Alas for her! Reveal her name!
Magic Mirror: Lips red as the rose. Hair black as ebony. Skin white as snow.
Queen: Snow White!
You can see how someone might interpret the highlighted line as the mirror revealing Snow-white’s real name. But I think a more natural interpretation is that the mirror is giving the queen a (not very difficult) riddle, in the usual manner of oracles.
Another version of the story that that uses all three of Snow-white’s attributes as a kind of epithet, is a 1982 retelling by Nancy Garden:
“Nothing can save you now, my beauty,” chortled the queen, “for I have used my strongest poison this time, and since you have swallowed it, you will never rise again. So, Skin-White-as-Snow, Lips-Red-as-Blood, Hair-Black-as-Ebony, what good is your beauty to you now? This time no one will ever be able to wake you up again.”
Nancy Garden (1982). Favourite Tales from Grimm, p. 15. New York: Four Winds.
Again, you can see how a reader of this passage in Garden’s version might conclude that “Skin-White-as-Snow, Lips-Red-as-Blood, Hair-Black-as-Ebony” is Snow-white’s real or unabbreviated name. This is an appealing kind of fairy-tale idea! But elsewhere in Garden’s version she is just called “Snow White”, so it makes as much, or more, sense to take the queen’s phrase here to be a sarcastic epithet that she made up on the spot.