How come it's mit? Would you assume a neuter noun had been mentioned before (Like "(Du har dit dyr og) jeg har mit") or could it be either in this case?
In this case, you would know that a neuter noun had been mentioned before, precisely because of the 'mit'. If it were a common noun, it would be 'min'. In fact, you only ever use possessive pronouns with something that is known or has already been mentioned, otherwise you'd use a possessive adjective plus a noun.
Of course, if you're translating from English, either would be fine, since you don't know the context.
I suppose it is as in Swedish. We use 'mit' and 'dit' when something is 'neutral', i.e. both if neuter noun or if talking 'in general', no noun implied. More like the English 'mine' and 'yours' .
To be quite honest, I am not 100 % sure why I do what I do when I speak my own language (Swedish), so I learn Danish here to see if I understand Swedish better - that's the mystery with 'native language'. To rephrase what I said above: you say: "Det er mit" or "Den er min" so it refers to the implied noun. Probably that's it. Though 'Det' feels more neutral, like English 'it'.
You get a much better understanding of the mechanics of your own language once you can compare it to the workings of another :)
See the comment by Balaur above (and just to add, mine is used if the noun you're referencing is in plural)
I have mine could also be jeg har min; correct? It depends on what that min/mit is referring to, right?
So, is it mit here because there is no subject to refer to afterwards? For example, in the sentence, "Jeg har mine æbler " mine is referring to the apples. When there is no word to refer to, does it just (for lack of knowledge I have for a technical term) default to mit?