Why does Montevideo has a specific translation in Taiwanese? Translation of Montevideo in most dictionaries appears to be: 蒙得維的亞 (Méng dé wéi de yà) but in Pleco's dictionary it also says that in Taiwan is used: 蒙特维多.

As far as I am concerned, in Taiwan it should be used the same translation but with traditional characters, why does it have another translation?
It is a fact that different countries and areas can have different translations or transliterations of a foreign word or name. It can only be resolved by the name's owner to give official translation or transliteration for other countries. 

Take Google for example, Google itself decided it would be '谷歌' in Chinese. That makes it official and universal in all countries and areas that use Chinese as their written language.

Another example  is the name of a Canadian city, Montreal. It was transliterated as 蒙特利尔 in China but Cantonese transliterated it as 滿地可.  Since the early Chinese immigrants to Canada were mostly Cantonese, 滿地可 is the accepted name for Montreal among Chinese Canadian. 

The Canadian government just transliterate Montreal as 蒙特利尔 in Mandarin and  滿地可 in Cantonese.