Measure Words and their English Analogue
I've been taught, however, that in Chinese, every noun has one and only one correct measure word associated to it, and in particular, you can't make the analogous distinction between the two cases in Chinese.


This is completely wrong. Measure words in Chinese do carry meaning, just like in English. 

Using the wine example, all the following are correct:


一瓶酒 (a bottle of wine)
一碗酒 (a bowl of wine)
一壶酒 (a kettle of wine)
一杯酒 (a glass of wine)
一滴酒 (a drop of wine)
一斤酒 (a jin (a Chinese unit, =0.5 kg) of wine)


Because wine is uncountable, the measure words are container names, or volume, or weight. 

For countable nouns, such as 'person', there are singular measure words and collective measure words (also known as group measure words):


一个人 (one person)
一群人 (a group of person)
一队人 (a team of person)
一大帮人 (a big gang of people)
一小撮人 (a small proportion of people (from a large group))


Even within singular measure words for the same noun, there are different focuses corresponding to different contexts, such as:


一堵墙 (literally 'a blockage of wall')
一面墙 (literally 'a surface of wall')


When you want to say the wall blocked the exit, you may choose 一堵墙挡住了去路, while expressing a wall-ful of photos, you may want to say 照片多得挂满了一面墙. Although in this case switching them is fine (still grammatical), it would become less consistent in inner semantics, hence less literary.