Ya know EyeseeCold brings up a pretty valid point!
To the average American who flippantly throw about their 'nationality' based on stereotypes and presumptions and little else, what really is the difference between Irish and Scottish? Often they know little about either, have visited neither and claim connections to both for amusement or pride without really knowing much about the place and people.
Really to such people there wouldn't be much difference on an emotional/cultural level if they were distantly related to anyone from any of the countries in the British isles.
I'm curious as to how something with such little impact on your life can be 'shocking'. I meaaann. Presumably you've never really interacted with Ireland or it's people and you aren't likely to do so with Scotland any time soon.
If all you can say is you share a small strand of probably mostly bred out DNA with a long dead Scotsman then does it really matter at all?
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Isn't the average American like Irish/English/Italian/Australian and Spanish?
And on a deeper level, quite a lot of the Scottish are Irish and vice versa, same melting pit! So if we're talking heritage-wise genetic traits since you obviously won't have any cultural traits, then really there is no difference. You're still a Celt.
Shit. Sorry I'm mostly rambling to myself here.