"put an article on the word" is this sentence awkward?

I do not know, of course, what your teacher was intending or looking for in the question.

However, I would make two suggestions.

First, 'Samsung' -- within quotes -- is a company's name. Samsung -- without the quotes -- is a company. This is the distinction between use and mention. In your sentence, you are referring to the company Samsung, not the name 'Samsung'. (You get the distinction right in the next sentence when you put quotes around 'article'.) Hence, the sentence, whether or not the article 'on' is appropriate, is poor.

Second, yes. Using 'on' in this way is awkward at best. The preferred construction is 'take an article'.

Rewriting your sentence, I think this is better:

'Samsung' is a company's name, so it does not take an article."

For an example, search for 'take an article' on this site:

https://www.oecd.org/about/publishing/OECD-Style-Guide-Third-Edition.pdf

I hope this is helpful!