"Jeg har min" is for a simple object and "Jeg har mine" is if you have more of a kind, if that makes sense? :)
Mine is for plural, so probably in this sentence they are referring to a plural subject.
This sentence doesn't make any sense, does it? At least in english it seems that you couldn't say it without referring to something already mentioned. How does that behave in danish?
"Hey, do you have your waterbottle?" "Yes, I have mine"
Although I think "Jeg har mine" should be correct too since we don't know what is being referred to.