In comparison to another phrase mentioned; another factor to consider: "factory" is not a proper noun, but a common one. On the other side, "America" is a proper noun. The country in question is always a proper name as well, and in English, proper names cannot be modified by an article or other determinants. Nor do they modify other words.
This carries over, in that we also say "American made" or "French made", rather than "America made" or "France made" when referring to the manufacture of items. We would say "France made a mistake" to refer to the government of France doing something, but not "France-made clothing". The other difference is that in "France made a mistake" it is a noun + action structure.
By understanding them as proper names it should become clear that the supposed alternative phrases must be ungrammatical. The word "America" refers to the country itself, while "American" refers to being of or related to the country called "America". Proper names are simply not adjectives or modifiers - they refer to a specific thing. The prase "factory made" uses a common noun, which can be used in this way because it isn't a specific reference. One exception here is that you might say "Dearborne made", but this exception exists because there is no general referential term meaning "from, of, or related to Dearborne"; ie. China -> Chinese, France -> French, Dearborne -> ____.