GRRM has never explained exactly what “the song of ice and fire” is, either in the books or in interviews. It’s one of the ongoing mysteries of the series, but of course is vital to it (being its title and all). There are many possible meanings for the concept, and it’s probable that several, if not all of them, may be true at the same time. These include:
- winter / summer (i.e. the balance of the world’s unusual seasons)
- hate / love
- death / life
- The Others (White Walkers) and their wights / the dragons, also obsidian (dragonglass) and possibly Valyrian steel (dragonsteel)
- The Great Other / R’hllor
- Lyanna / Rhaegar (also Stark/Targaryen in general)
- Jon / Dany
Relevant quotes within the books:
“We swear it by ice and fire,” they finished together. (Jojen and Meera, ACOK)
“He is the prince that was promised, and his is the song of ice and fire.” (Rhaegar, ACOK)
“My lord father told me about mountains, but I never saw one till now. I love them more than I can say.”
Bran made a face at her. “But you just said you hated them.”
“Why can’t it be both?” Meera reached up to pinch his nose.
“Because they’re different,” he insisted. “Like night and day, or ice and fire.”
“If ice can burn,” said Jojen in his solemn voice, “then love and hate can mate.” (ASOS)
“The war has been waged since time began, and before it is done, all men must choose where they will stand. On one side is R’hllor, the Lord of Light, the Heart of Fire, the God of Flame and Shadow. Against him stands the Great Other whose name may not be spoken, the Lord of Darkness, the Soul of Ice, the God of Night and Terror. Ours is not a choice between Baratheon and Lannister, between Greyjoy and Stark. It is death we choose, or life. Darkness, or light.” (Melisandre, ASOS)
Another thing to keep and mind is Robert Frost’s “Fire and Ice” poem, which GRRM has mentioned as an inspiration:
Why your saga is called A Song of Ice and Fire, because of the Wall and the dragons or is something more beyond that?
Oh! That’s the obvious thing but yes, there’s more. People say I was influenced by Robert Frost’s poem, and of course I was, I mean… Fire is love, fire is passion, fire is sexual ardor and all of these things. Ice is betrayal, ice is revenge, ice is… you know, that kind of cold inhumanity and all that stuff is being played out in the books.
So, for Jon and Dany, while they definitely have aspects of ice and fire, that’s still on the level of fan interpretation / analysis rather than absolute textual truth. As you said, it’s a very popular and accepted interpretation, but it has yet to be confirmed by GRRM in any way.
Regarding Sansa and Arya, there was a fine meta that asoiafuniversity reblogged recently that agreed with you — that Sansa is ice and Arya is fire. Personally, if I were to put them on the scale (and I do), I prefer the opposite interpretation. With the concepts laid down by Jojen, Melisandre, and Robert Frost’s poem, where fire is love and passion and ice is hatred and revenge, the ice/fire aspects within the girls seem obvious to me.
Sansa, fiery-haired, is all love and romance (albeit mildly because of her youth, but it’s interesting that she is the most paired character in fanfic nevertheless), who cannot escape plots centered on her marriage status. Arya, the most “Stark” of her siblings save Jon, is motivated by hate and revenge, reciting a nightly prayer of vengeance, becoming a cold-blooded killer.
However, that meta has a definite point: that Sansa is often icy calm and collected, whereas Arya’s passion and burning hate shows the fire within her. So I think, in truth, the girls show aspects of both ice and fire — and that may be the true meaning of the series title when applied to them, if anything.