I have never heard "brothership" or "daughtership" as a native English speaker. "Brotherhood" is usually used figuratively, as in the phrase "the brotherhood of man." The suffixes -ship and -hood often mean the same thing, but they aren't used interchangeably; for example, it's parenthood, brotherhood, sisterhood, fatherhood, motherhood (the latter two describing the state of being a parent), but relationship.

Only the word "brotherhood" is in use. There are NO such words as brothership, sistership, or sisterhood.

I don't see a difference between brotherhood/brothership, etc they are the same, also you can say 'brethern" daughterhood/ship are the same sisterhood/ship

I agree with Ashley and Lee. 'Brothership' and 'sistership' and 'daughtership' do not exist. We have 'brotherhood' and, less commonly 'sisterhood' to refer to a community or sense of community. And - rather confusingly - we have 'childhood' and 'adulthood', in which the 'hood' suffix has a different meaning. 'Childhood' and 'adulthood' refer to these periods of our lives. And, as Ashley says, we also have 'motherhood', 'fatherhood' and 'parenthood' to refer to these states. Words ending in 'ship' connected with relationships are 'relationship' itself and 'friendship'.

"ship" endings are very seldom used, and may sound funny to many native speakers, but yes they do exist, but are probably obsolete or archaic.