I'd also beg to differ on your point that Canadians have no ancestry of invasion - Canadians descend primarily from France and the UK (with a little German as well), and it would be a tad outrageous to pretend these nations haven't endured bloody and violent invasions throughout history.
And grandparent's point wasn't that Canadian's don't have a cultural heritage of invasions and battles, but rather that the Mongol invasion was somewhat different from most of the things going on in Western Europe (e.g. almost all major cities were sacked; casualties were at levels not to be seen in Western Europe until World War 2) and had widespread and long-ranging effects on society that made it qualitatively different from Western Europe. Second paragraph of the "Influence of the Mongol invasion on Rus' society" section of the cited Wikipedia article is relevant here.
The very notion of the American dream is codified on New York's main landmark: "give me your poor huddled masses" etc. The story of America is necessarily the story of many populations coming together to create something new. The Irish may have the old country to look back on, for example, but their original cultural heritage has been superseded by that of America itself.
Russia has no such story; along with its disparate satellites (Ukraine, Belarus and so on) it has a long history of single peoples mostly staying still, or emigrating. The idea of a "Russian people" with a common cultural heritage has been a fairly constant concept for hundreds of years, whereas "the American people" is a very new one, historically.
It's not something I'd dismiss out of hand. For example, England was invaded by the Normans over 900 years ago, and the results of that invasion still affect everyone's everyday life today. Or consider that some cultural entities can be very long-lasting, such as religions, languages and writing systems.