Is there a Chinese saying similar to “Once two men share a woman, they’re brothers.”
I believe it is a Cantonese term.

Traditionally, a man would call the husband of his wife's older sister 襟兄; and that 襟兄 would in term referring the man as 襟弟. They were 襟兄弟 (lapels brothers) or 老襟 (fellow lapels) to each other.

襟兄弟 originally describe the relationship of two men married a pair of sisters. Somehow, people began to use this term to refer to the relationship of different men married the same woman in different time. 

To avoid confusion and being joked at, people stopped calling each other 襟兄 or 襟弟 and started calling each other 姊夫 (older sister's husband) or 妹夫 (younger sister's husband) instead. 

Outsiders in occasions would still refer them as 襟兄弟, with the understanding of the term also has a second meaning that is derogatory.

Nowadays, calling the men who have sexual relationship, (married or not) with the same woman in different time (or worse, at the same time) as 襟兄弟 (lapels brothers) or 老襟 (fellow lapels) had became the main definition, while the original definition of describing the relationship between two brothers-in-law had became secondary.

a Chinese reference

a Cantonese reference


  There's also a term 同靴 (in a same boot) or 靴兄弟 (boot brothers) 
  refers to men who have sex with a same prostitute
  https://zhidao.baidu.com/question/62611723.html 
  
  旧时称同嫖一妓者。《官场现形记》第三二回：“幸喜他平日，也常到 钓鱼巷 走走，与 余荩臣 有同靴之谊。”
  
  同靴之谊 means "relationship between fellow prostitute clients".  It also means "relationship between two men who have sex with a same woman" nowadays.