Huh, I thought I was the only one who couldn't do the hoodie/jacket/car combo!
Anyway, this is an interesting question and I have no clue what the answer is. I was always claustrophobic, my mom would tell stories of me not being able to sleep on bottom bunks and not wanting to play in boxes like normal kids as young as 2-3. That seems like a really young age to "pick up" on a phobia. Especially since I didn't learn my mom was claustrophobic until I was in middle school when she had to get an MRI and was freaking out about it. My brother isn't claustrophobic but is freaked out by spiders (another one from my mom) but I love spiders. I am afraid of heights but neither of my parents are.
So... good question. I guess IMO it's 50/50. We might just naturally have fears, whether due to nurture or just our own mind playing with us, or maybe they are genetic. I'd be interesting in learning more when I'm not so sick/tired.