This could mean several things.
The speaker is a cow. The speaker is a mother who is allowing her child to breastfeed. The speaker is a mother who is allowing someone else's child to breastfeed. The speaker has legal claim to the milk that the girl is drinking.
Which do you think? Are there any other possible situations that I missed?
I didn't even think of breast feeding. I thought it was like, I have a carton of milk in the fridge which everyone understands to be mine, but the girl is drinking from it anyway, lacking respect for my property. lol. The idea of breast feeding didn't even occur to me until I saw this comment. x3
Since no one answered though it might be late, no, there is no difference whether you want to say that girl is drinking milk right now or she does it occasionally on regular basis. It would be still Девочка пьет моё молоко. You have to guess by context. But sometimes such words like "always" or pointing any specific period of time by the speaker can help. For example: Девочка постоянно пьет моё молоко (Girl constantly drinks my milk).