as well as 개 – 'things' in general - there are lots of specific counter words in Korean, e.g.
곡 – pieces of music
명 – people
장 – sheets of paper
In most cases, I think using 개 rather than the correct counter would be grammatically incorrect.
Are there any/many counter words that are sufficiently obscure that the average adult Korean speaker might not know them? Or do they represent a limited and fairly well-known set?