Why west and wine vessel/10 th terrestrial branch are similar?
The first thing to point out is that both west for 「西」 and 10th terrestrial branch for 「酉」 are both rebuses or phonetic loans. This means that the shape of the character does not have anything to do with the meaning of the word that it represents.
The character 「西」 represented a word which sounded similar or sounded the same to an existing word meaning west, therefore it was also used for the word west, and the same idea goes for 「酉」.





時期
「西」
「酉」




商
甲　前5.13.2合集34340
甲　乙9021合集20856


西周
金　伯⿹戈冬簋集成4115
金　三年壺集成9726


戰國
晉・璽印　3966古璽彙編
燕・璽印　3447古璽彙編


秦
簡　日書乙種163睡虎地秦簡
簡　日書乙種225睡虎地秦簡


西漢
隸　二年酒鋗
隸　新承水槃


東漢
隸　史晨碑
隸　韓仁銘


—
楷　
楷　




The similarity in appearance between 「西」 and 「酉」 is nothing but a long history of shape stylisation.
「西」 originally depicted a bird's nest, and this word is now written as 「棲」 (Baxter-Sagart OC: /*s-nˤər/). The character 「西」 was borrowed to mean west (also /*s-nˤər/) since the earliest traceable inscriptions, but this borrowed usage is a phonetic loan.
「酉」 originally depicted a container for storing alcohol. This was very early on extended to mean wine, alcohol in general, now written 「酒」 (/*tsuʔ/). An abundance of characters containing 「酉」 or a derivative of 「酉」 as a semantic component have something to do with alcohol or fermentation:

「醬」 fermented soybean sauce
「尊」 ceremonial wine vessel &gt; wine ritual &gt; respect
「釀」 to ferment
「醉」 to be drunk

The character 「酉」 was also very early on used for one of the terrestrial branches (/*N-ruʔ/), but this usage is also a phonetic loan.

See e.g. 《甲骨文合集》37986 for a record of the calendar days inscribed on an oracle bone. In this sample, 「酉」 is the bottom-most character on the right-most column.



References:

季旭昇《說文新證》
小學堂字形演變
國學大師

郭沫若《甲骨文合集》
中國社會科學院考古研究所《殷周金文集成》