Calling each other brother and sisters is not just an Islamic thing but Christian one also. In addition it is also a cultural thing. Chritains call each other brothers in the same way as Muslims do. They might have a better explanation for this. According to this site, which make sense
Because we are all Gods children and that makes us siblings, so to speak.
Because we are all brothers and sisters in god
Keeping in view that we are all children of Adam and Eve (from Islamic point of view), we are in a way brother and sisters.
It seems strange that you get married to someone who you referred to
(in the passed) as your brother or sister?
This is a valid concern. But usually when you call someone sister, you show her respect and that you do not have any awkward feeling towards here. That makes the communication far more easy then staring with something awkward, which may not be appropriate.
Speaking of the such verbal relationship, the prophet (PBUH) married wife of his adopted son Zaid bin Harithath to stress that it was only a verbal relationship and not a blood relationship, therefore it was no longer valid. So this is a little bit contradiction here. While calling a woman a sister does show sign of respect but then she can never become your real sister. You can marry her at some point if you want to, which is awkward, at least to me.