What is meant by clothing that is shared between men and women, and is it permissible for me to wear my brother's black T-shirt with a rectangular stripe of shapes on it while at home?

It is impermissible for a woman to wear men's clothing, and for a man to wear women's clothing, and this is considered among the major sins. This prohibition applies to garments that are conventionally characteristic of each gender. As for shared garments that are not exclusive to a specific gender, both genders are permitted to wear them. Determining what is characteristic of each gender is based on the prevailing custom in each country. Therefore, if the "T-shirt" mentioned in the question is not specific to men according to the custom in the questioner's country, she is permitted to wear it at home, provided that she does not adorn herself or expose her 'awrah when going out.