There is nothing special about this word. The distinction is very similar (if not the same) to the English countable/uncountable nouns. "Вещь", in its plain literal meaning, is obviously countable, and so should be plural.
Occasionally, you can have both, with different meaning:
Of course, in English, you also use "few" vs. "little" in these cases; in Russian, the word is the same, but the opposition still exists.
I suspect the source of your confusion comes from this thing. The thing is, the word "thing" is (ab)used in English for many things. Not so (much) in Russian: "вещь" is mostly literally "an item". It can be used like in English, as a placeholder subject, but this is more often than not a poor style; often an calque from English. Usually you want to avoid it.
There are more things in heaven and earth, Horatio,
Than are dreamt of in our philosophy.
Есть многое на свете, друг Горацио,
что и не снилось нашим мудрецам
Still, even in such cases "thing" is countable. But the whole expression "many things" = "многое" is not.
Many uncountable nouns still have a plural form, just like in English, and so are listed in your dictionary. They usually represent classes of things of abstract concepts:
воды (pl.) ~ waters (though usage is a bit different)
времена ~ times (very similar usage, e.g. "those times" "те времена")