Dobrze! If "twoja kobieta" is having breakfast, at least "moi mężczyźni" have wine :) :D
Twój/twoja/twoje/twoi are for singular you -Larry and nobody else
Wasz/wasza/wasze/wasi are for plural you (you all) 2+ people
Mój Twój nasz wasz are for describing masculine nouns
Moja Twoja wasza (last letter is a) are for describing feminine nouns
Twoje wasze nasze (last letter is e ) are for describing neuter nouns and most plural words
Moi twoi nasi wasi (last letter i) are for describing plural nouns that describe men
Unless, perhaps, it is stated affectionately. "My man kindly did all my errands for me today. He's the best."
isnt it like an analogue to 'having some wine', i.e. to drink some wine? Why does it suddenly means 'possesing a wine'?
In Polish "mieć" is always about possesion. sometimes literal sometimes metaphorical like have a dream, have thirst,
Frankly, I'd use it only when a woman refers to her husband and her sons as "my men".
Is that common usage in Polish? I'm Spanish, and in Spanish it is very unusual. The only situation in which "my men" ("mis hombres") would be used in Spanish is when a military leader is talking about his troops, and even then it would be very rare.
"moi mężczyźni" is really rather strange to me, so I'd probably really limit it to "my husband and my sons".
"moi ludzie" makes more sense and it could be the military context you mentioned, it could also simply be a boss talking about his employees.