First, you must realize that the name of AC Milan deliberately shows the name of the city in English, instead of Italian (Milano). Italian fans leave the word untranslated. Simply, the name of the club is “Milan”, and it is an association devoted to the practice of football, hence the words “Associazione Calcio” that accompany the club’s name.
However, the name of their nerazzurri neighbors is FC Internazionale, Inter for short, a club which happens to be also based in Milan, Italy....
...It is different, however, in Manchester, where I think it would be incorrect to refer to any of their two main clubs simply as “Manchester”, and you would need to especify whether you are talking about the “United” or “the City” -if I am wrong here, happy to be corrected.