I think it's a mix.
I also think that once you experience winning, it can be addictive. My brother for example, as a toddler was AMAZING at everything, could talk in sentences by the time he turned 1, rode a bike without stabilisers before 2, could write full stories by 3 etc. He was 'the best' at everything, he won all the swimming races, the running races, the who can collect the most ball games. This made him competitive. And as his peers caught up (by the time he was 10 he was bright, but a normal kid again) he didn't cope well with losing.
I am the most uncompetative person you will ever meet. Maybe that's because my brother always won, so I got used to it and didn't care. There is less than 2 years between us.
My younger brother, 10 years younger so pretty much wasn't brought up with us, we were at boarding school, then university, then left home, we never even really lived with him. He is SO competitive, just like my older brother. But he is dyslexic and dyspraxic, he's not good at English, or math, or sport. But my god does he work hard to get there. That's got to be nature. No reason for it, my parents are incredibly non-competitive, just like me.