I'm not a native English speaker, but I'm trying to explain it from my limited knowledge of syntax.
If you choose to use "has not" structure, the "has" here is no longer the main verb, but the auxiliary verb, a verb modifying the main verb (there should be better explanation or description, but that's the best I can come up with).
For example, "I have been studying Italian." "Have" is just an auxiliary verb here, showing that it is an ongoing action which has lasted for some time. What is the ongoing action? The main verb, "be studying". Or, "I have been to Thailand." The "have" here is also an auxiliary verb showing that it is a finished action ended just a while ago, and "been to" indicates the action, hence the main verb. So I guess it's fair to say when "have" is used as an auxiliary verb, it's more about indicating the process of the action and the time, rather than showing possession as it does when acting as a main verb with dummy "do" as the auxiliary verb.
In this sentence, "Lui non ha le scarpe degli uomini." "Ha" is clearly showing possession, one does not have/get/obtain/etc. the shoes of the men. So, the better and more literal translation would be "He does not have the shoes of the men." or the one suggested by Duo to me, "He does not have the men's shoes."
Yet again, if you really want to stick to the has not structure (which though I personally think is less literal and less correct), the one suggested to Olga by Duo, "He has not got the men's shoes" will do too.
Hope that helps :)