I think this is a false dichotomy, really. Both nature and nurture are extremely significant!
I can see in both my children similar personality traits to me and DH, which comes down on the "nurture" side.
DS is musical, impatient, focussed, competitive, with lots of oomph. He can't eat a few sweets and save the rest, he has to wolf the lot in five secs flat. Like me.
DD is an artist, gentle, observant, slow, methodical, has no competitive gene and is a measured person. She will have one or two sweets and save the rest. Like DH.
However, I think the way we interact with others relationally is much more down to nurture and replication of childhood relationships/dynamics. Your attachment style (secure, anxious, avoidant, disorganised) will derive not from your genes but from your relationships with significant carers in infancy.
Finally a word on "we had exactly the same upbringing" so differences MUST be nurture. Hm. Of course two children being raised in the same house will have many similarities in their experiences - same parents, same physical environment, same parental philosophy.
But beyond that each child has an individual experience, shaped by birth order and what is going on in the parents lives as time marches on. Superficially you would say my sister and I had the "same" upbringing - same house, same village, same activities, same school, same parents, same parental attitudes, same holidays.
Actually, my sister and I had quite different upbringings. I was the eldest by 7 years, born to young and optimistic parents, I was an only child, the sole focus of parental love and expectation, etc etc. My sister followed my brother who died neonatally when I was four, and she was born to quite different parents who were bereaved, anxious and who had suddenly realised very immediately how precarious existence can be. She also had a precocious and bossy elder sister. For her it was like growing up as a very small person with three very big people in the family looking down on her. Whereas for me it was like growing up on a stage all of my own, with two adoring adults in the audience.