I'm not quite sure what you're saying here. "I've" most certainly does denote possession. "I've (or I have - depending on emphasis and context) an exam in the morning", "I've an appointment at the doctor in the morning" etc are both perfectly fine.
I think the usage "a training" sounds wrong: I've never heard that (but it may be used in some places), it would either be "a training session" or just "training".
I think praytherosary may be right about this. For the most part you will never hear someone say "I've + noun" in the US, but I'm not sure it's actually a solecism. Though, it's so totally nonexistent where I'm from, that I can see why Patrick would argue that.
Are there any other native speakers of a different dialect of English for which this construction is common?