It is because you are now saying "I also do not have one sister", implying that you may have none, or more than one.

I can imagine that it would be like saying "I also don't have (exactly) ONE younger sister. (But maybe two, maybe three.)" Weird... Here you want to say that you have no sister (in general). I guess you can imagine it as plural if you want to translate it more directly: "I also don't have (younger) sisters".

I guess perhaps it would be the difference between saying "a sister" vs. "one sister"? Still, I don't think the meaning is that different...

没有 means don't have, and 不 means no. so it would be weird if you said 我不妹妹 because it would mean I no younger sister

No, you can't use 不 with 有, in that case 沒 is always used instead.

I was wondering about that 不