This is a bit meaningless. D&D isn't GoT, and GoT isn't D&D.
But, we can guess that very large numbers of the new D&D players out there know GoT, and may find the framing familiar and attractive. Entire genres of fiction are successfully based on copying themes, so we can hardly say that picking up a theme for one adventure is somehow a lousy idea.
And, by the same argument structure D&D isn't modern reality either, so there's no reason for it to match reality. And the whole idea that the 2-year winter doesn't make sense is based on expectations of current reality.
Our current time has us using a food and resource stream that is "just enough, just in time", enabled by modern shipping practices. However, back in the day, folks who depended on local weather conditions for their livelihoods generally kept large stores of food locally, because any particular year could have a drought, or other weather pattern that devastated food sources. Well-prepared castles filled with people could in some cases last
years of siege.
In the Bible, forewarned by Joseph, the Pharaoh was able to prepare his land for
seven years of drought and famine, with enough to spare to take refugees. So it isn't like legend somehow lacks precedents, such that this is outlandish in our stories.
So, for this to make sense... the people plan for bad years! One bad year went by, and they didn't sweat it much, as they were prepared. The
second bad year has gone by, and now they are worried.