What does the 月 (moon/month) in the Taiwanese MRT platform's 月台 refer to?
None of the references is very authoritative, but all of the search results agree with each other, that 月台 for 'railway platform' is an extended meaning from its original meaning 'moon-watching balcony' which refers to a platform that protrudes from a building.

月台 （建筑样式） 编辑 讨论
在古时建筑上，正房、正殿突出连着前阶的平台叫“月台”，月台是该建筑物的基础，也是它的组成部分。由于此类平台宽敞而通透，一般前无遮拦，故是看月亮的好地方，也就成了赏月之台。而现代的月台通常指进入火车站后方便旅客上火车的一段与火车车门踏步平行的平台。

https://baike.baidu.com/item/%E6%9C%88%E5%8F%B0/871716

https://www.guokr.com/question/543681/

https://www.rrb.gov.tw/07211.aspx?id=652

https://www.zhihu.com/question/29310884