My thoughts and beliefs have been greatly shaped by my dad, and I'm positive my dad is who is is at least partially because of his relationship with his grandfather. Carry that backward, and I'm only three degrees removed from a soldier in the Civil War -- it's not impossible to think, completely independent of genetic inheritance, that some of my personality quirks or principles worked their way straight to me from someone who was born in the 1840s.
But in this case, not actually as far away as you are thinking, because of something called pedigree collapse. For example, the second row in your family tree, your grandparents, has 4 people, 2^2. The 20th row, only about 500 years ago, has mathematically 2^20 = 1,048,576. In reality far fewer, because the same people fill multiple spots. Only 1,000 years ago, 2^40 = 1,099,511,627,776.
A trillion ancestors 1,000 years ago? Doubtful.
So you can see, this exponential explosion is not an accurate way of thinking about ancestry.
Pedigree collapse within a certain area is what leads to the various human races. The distinction of race is simply the description that someone is descended from some localized subset of the global population.
You're far more closely related to your g^40-parents, than the g^40-parents of someone half-way round the world because your g^20-parents are more closely related to your g^40 parents. And so it goes up and down the tree.
So yeah, maybe only ~3% at a certain level, but that is as closely related in terms of match-able DNA sequences that it is possible to be at that distance in time. You're still very closely related in terms of the overall DNA "color palette."
And it is likely somewhat more than 3%.
Many Ashkenazi Jews have a 'genetic relation' that is in fact higher than their 'genealogical relation' would suggest because there has been such a high degree of intermarriage among a small population over the years. This is the case for all races, just maybe more pronounced in Jews because of the smaller population.
There's something sad about such a statement, and it epitomizes a certain cause of present day problems in society.
Are your genes the only things that "has to do with you"?
What about the built environment and society in which you exist with all of it's inventions and innovations from a strictly primal state of affairs?
Who built that society and environment around you? If you're like most people, your ancestors.
What aspects of your ancestors guided their building and structuring of society? Their genes. Your ancestors have very much to do with you, and you have much to be grateful to them for.
Edit:
p.s. As a corollary addendum, if your ancestors have nothing to do with you, what of your descendants? Why bother leaving an orderly and healthily functioning world to people "who have nothing to do with you"?
Safe to say, that we should pray for the well-being of future generations that most people today don't think this way, and be thankful that most people in the past didn't think this way.
Is that a population level statistic?
There are maybe 2-3 cross-over events per chromosome per generation in humans, e.g. https://biology.stackexchange.com/questions/19005/how-many-r... .
You share about 0.5^n percent of your DNA with each member of generation n, where n = 1 are your parents, n = 2 your grandparents, and so on. The explanation that was given to me is that each of your parents contributes 50% to your genes. Depending on which of their parents' genes they gave you, you share anywhere from 0 to 50% of your DNA with each of your grandparents, but for simplicity's sake let's call it 25%.
The point is, most of the people we're trying to find out more about through ancestry websites and DNA tests are barely related to you. Searching for your great, great, great grandfather from the old country? Fine, you only share about 3% of your DNA with him.