It is different. "是的" means "That's correct"; So if someone ask me "you don't smoke?" and I answer "是的" that means I don't smoke, which is actually what "no" means in standard English.
I don't think there's much to be learned by trying to examine this situation in English. Affirmative answers to negative questions are a source of frequent confusion among native speakers, suggesting that the rules of English are not clear on this point.
If you said to me 你不是犹太人吧？, and I responded 是的, what would you think?
However, I do see Chinese making mistakes in English and English speakers making mistakes in Chinese and I am not 100% free of the guilt.
In the question "你不是犹太人吧？" there is already a "是“ in it, so a native speaker would use "对“ or "没错“ instead of "是的". They all means "correct" but by avoiding using "是" again, the chance of misunderstanding is minimized.
> There are clear rules in English and Chinese.
> They all mean "correct" but by avoiding using "是" again, the chance of misunderstanding is minimized.
If the rules were clear, there would be no misunderstanding.
你去不去商店吗? (Are you going to the store?)
The way to answer is either going (去) or not going (不去). There are a lot of phrase for "that's right" or similar, but that's not the same as "yes."
Mandarin has no word directly mapping to "yes."
Your example is not in the form of a yes-or-no question[1], so it's not surprising or interesting that the answer can't be "yes". We see exactly the same phenomenon in English:
Q: Are you going to the store or staying home?
A: *Yes
[1] Admittedly, the presence of 吗 makes this claim weaker.
(I started typing 你去不去商店 and changed part-way)