In my opinion,
The Ice Dragon is a world apart from Westeros. There are similarities sure, but there are also dissimilarities.
Similarities consist of dragons, kingdoms, war, summers, winters, building snow castles, Beth, and Hal. These all show the humanity of the world. Cruelty, toil, nationalism, and games. Adara and her family come to life during the seasons, the trials, the heartbreaks, and joys of their lives. The names Beth and Hal occur. in ASOIAF, once and twice respectively... plus six other names that begin with Beth and seven others that begin with Hal.
Differences... ice lizards, dragon riders are not nobles, winters seem to come at least once a year or even more frequently, John, and the lack of the Wall. We've gotten a good amount of POV from Arya, Bran, Jon, Eddard, Catelyn, Sam, Tyrion, Theon, Asha, Will, and Varamyr about their lives in the North... Bran's story shows the supernatural while Jon's especially dealt with his life among the Wildlings and none of their accounts reveal the existence of ice lizards.
In ASOIAF, dragons seem to be the exclusive property of the Targaryen royal family.
The World of Ice and Fire says that dragons were only used by the noblest of Valyrian families.
The winters in TID come frequently, but maybe this is just the perception of Adara.
TID also lacks any mention of the Wall. This could place the story well before the raising of the Wall, i.e. at least eight thousand years before Jon and Dany... but this explanation distances Adara from the Starks and all the people of Westeros whom we know.
Adara's father is John. John does not appear at all in ASOIAF, but the name Jon does.
Twenty different characters bear this name... and seven others begin with Jon, while none begin with Joh.
From what we know right now,
The Ice Dragon does not fit into ASOIAF... but GRRM could easily rememdy that if he chose.
The Ice Dragon could be placed years and years before the raising of the Wall or even after ASOIAF. Some readers are convinced that the Targaryens put a dragon into the Wall.
Placing
The Ice Dragon over eight thousand years before ASOIAF or hundreds of year after, distances the story so much that it is effectively a different world. I'll just continue to think of it as a stand alone story that influenced GRRM's epic.
