Is it common to refer to cousins as "cousin-brothers"? Most Chinese people I know don't have siblings. They refer to their cousin as "cousin brother / sister".  

Is this common?

If so, is this a modern change, perhaps reflecting the one-child policy?
It's not really clear to me what you mean with "cousin brother / sister". Do you mean for example 表姐姐姐 or a 表姐 simply called 姐姐.

To be complete:
表哥 or 表兄 is the older male cousin via female line. 表弟 is the younger male cousin via female line. Similar you have 表姐 and 表妹. Via the male line you similarly have 堂哥 or 堂兄, 堂弟, 堂姐 and 堂妹.

These words are not new.

Nowadays a lot of only children want a 哥哥 of a 姐姐 to "look after them" or to have somebody to whom to can go to when one has some questions or problems and they search for them (for example in the family or a good friend), and afterwards always call them 哥哥/姐姐. Also cousins are often considered as brothers and sisters and are often simply called 哥哥, 弟弟, 姐姐 and 妹妹.