Other forms: brotherhoods
The relationship between a boy or man and his male siblings is known as brotherhood. Feeling that kind of connection with other people, even if you're not related, is also a kind of brotherhood.
This word is limited to the male gender; in families, brotherhood is the relationship between brothers. To include all of your siblings, you can use the term siblinghood. Members of fraternal organizations — clubs or societies of men with common interests or jobs — also feel a sense of brotherhood. A more inclusive name, one that would allow for all genders, might be kinship, alliance, or association.