False friends are word that look similar between languages so that you might be tempted to use them with a meaning they only have in a different language. They are more a language learning/psycholinguistics concept.

False cognates are an issue of historical linguistics. They are words that look like they are etymologically related because of similar meaning and form, but have different origins.

False friends are an issue between languages. False cognates may occur within a language, for example isle (from Latin insla via middle French isle) and island (from old English iegland) have virtually no etymological connection although they look very similar and are synonymous.

Your teacher seems to have confused these concepts because magazine (both the French an English words, which are certainly not false friends to each others!) in fact is etymologically related to the French magasin and is certainly not a false cognate!

False cognates and False friends

• False cognates are words that are similar in their modern forms despite having different etymologies. This is regardless of whether the modern meanings are similar.
• False friends are words that are similar in their modern forms despite having different modern meanings. This is regardless of whether the words are etymologically connected.

(Part of an extensive answer that was given on Linguistics Stackexchange.)

Magasin and magazine are in no way false cognates because they have the same etymology (English magazine comes from French magasin).

You could say that magasin and magazine are false friends because they look the same, they could be pronounced the same (at least by an English native who mispronounces French) but they have different meanings.

Magasin and Magazine

French magasin came into French in the 14th century, from Arabic maḫzan meaning a warehouse. The word evolved into meaning as well the contents of the warehouse/store.

The English borrowed the word around the end of the 16th century. The word was also used in English¹ in the title of books with the sense of A storehouse of information on a specified subject².

It was first used in English with the sense of a periodical publication containing articles by various contributors for general readers with the spelling magazine by then more or less stable, in 1731 for the title of the Gentleman’s magazine
And the word came back into French as magazine by the end of the 18th century first to designate English magazines, and then any type of magazine.

_{1 I do not think it ever did in French.
2 Ref. OED.}