In English, we just use certain family relationships as names or terms of address while others we don't. Mother, Father, Grandmother, Grandfather (and related words) are all universally used as names / terms of address. Some relationships, such as aunt and uncle, may be in certain dialects, but are not commonly. Brother, sister, niece, nephew, etc. are just relationships. We don't call the people who have that relationship to us by that name. We use their given names. Only the words for older generations are used as names / terms of address. Actually, in your example, it would actually be much more common for both mother and brother to have my. It is very common to call your mother 'Mother' when addressing her. It is less common to do so when talking about her.

I think you've overlooked a crucial point. The writer is not talking to his mother or brother, they are talking to someone else. So MY is needed to identify who they are talking about. If they were talking to another family member (they father, for example), they would probably just use the person's name. Also, the others are correct, using BROTHER or SISTER as a form of address or substitute for a name is not generally done in modern English.

Yes, you’ve noticed an inconsistency in the original sentence! In English, when referring to family members in a general or narrative way, we usually say “my mother” rather than just “Mother.” However, if we are using “Mother” as a proper noun (like a name), then it’s fine to omit “my.” The issue in your sentence is that “Mother” is treated like a name, but “my brother” is not. This creates an inconsistency. Your revision: “Mother said that brother looked dignified in his new uniform.” This is not quite right because “brother” is not commonly used as a proper noun in English. Unlike “Mother” or “Father,” we don’t usually capitalize and use “Brother” as a standalone name. Best corrections: 1. “Mother said that my brother looked dignified in his new uniform.” (Most natural and correct in standard English.) 2. “My mother said that my brother looked dignified in his new uniform.” (If “Mother” is not being used as a proper noun.) Tip: If you’re treating “Mother” like a name, keep it capitalized and don’t use “my.” But for other family members, like “brother” or “sister,” it’s better to use “my brother” or “my sister” unless you’re in a very specific literary or cultural context.