I wish they would switch it so just "e menha garrafa" is correct then... I've never used that structure in my family's house. We are European Portuguese though
so using "a minha" is like saying "it's MY bottle" and just minha would be "it's my BOTTLE" ? is that the feeling for the a/o with possesives in general?
In this case it would be "my", not "mine". In English, "my" is the possessive adjective (it is my bottle) and "mine" is the possessive pronoun (the bottle is mine).