In one of the lesson notes it is said that есть can be omitted if the actual existence/possession of the object is not the point of the sentence. If you said "You have beautiful eyes", you would omit есть because the fact that she owns eyes is not the point, it is already known. The point is the adjective you attributed to her eyes, "beautiful". In this sentence, есть is omitted for the same reason. You are pointing out the adjective, моё.
Imagine a woman buying food in a supermarket. She brings a carton of milk to the register and lines up. Suddenly she remembers that she needs to buy something else. She asks the man who is next in line to hold the milk for her while she is running to fetch other food. When she is back, she puts her purchases on the belt and tells the clerk that she's also got milk. The clerk asks, "Where is your milk?", and the woman goes, "Мое молоко у мужчины" and waves at the man behind her. Such things happen all the time in Russian supermarkets.
With lots of nouns whose dictionary form ends in -a, the nominative plural coincides with their genitive singular. Anything that follows the preposition "у" (think of SHEL in Hebrew) is in the genitive case, hence "мужчины" in "у мужчины" is singular. In fact, it is one out 5 singular forms of the word мужчина. "у мужчины" means "at the man's/some man's disposal" and is normally rendered in English as "the man has" / "some man has".
Thank you, I just don't get how мой брат у мамы and Моё молоко у мужчины has such a different meaning, when the structure of the sentences seems a lot alike! :/ Or are these the kind of sentences that could mean both things, you should just think of how it sounds more normal?
Oh, because before it is an "у", whenever a noun comes after "y" and it ends with an "а", the ending"а" will change to "ы". Long long time ago, russian people said that way. Recently Russian people have said that way. In the future, they will say that way. So мы с тобой will say that way.