Why the sentence is " I am the short one in my family."
Not I am the shortest one in my family."
This is because there is a difference between those two sentences. If a person says, “I am the shortest one in my family,” it conveys the impression that there are other short family members but that he is the shortest one. On the other hand, when a person says, “I am the short one in my family,” this conveys the impression that his other family members are all either average height or even tall, so he is the only short family member.
It is like a person saying, “I am the chef in my family”; it conveys the impression that he is the only chef in his family.
To be honest I would say I am the smallest one in the family or I am the shortest one in the family. I could be short, or tall, but everyone else is taller.
If I say I am the short one in the family, that suggests both that I am the shortest AND I am short.
“I am the short one in the family.”
In German we would use such a phrase, too. Besides what has been said above it also conveys the impressions that the shortness of the person is an attribute directly connected to him (within the family, at least). So maybe even everyone in the family calls him “short one”. Whether he is objectively short might not even matter. (Maybe it is even the opposite and the whole phrase is meant ironically.)
English is very similar to German, so I would interpret the sentence in the same way in English, too.