Ambiguity in voltage measurement
So what does one do to avoid such ambiguity?



You can include a reference ("ground") symbol in your schematic. Then when you say you measure v volts at some node "A", we know what you really mean is that the voltage between "A" and the reference node is v.
You can clearly say which nodes you measured the voltage between. For example, "I measured v volts between nodes 'A' and 'B'" (but now you'll have to be careful to communicate whether you mean 'A' was v volts above 'B' or vice versa).



  And what does a manufacturer mean when he says a supply gives a 12V or 15V output? 


They mean there will be 12 or 15 V between the two output terminals, or between one output terminal and a designated ground/reference/return terminal.


  For if I change the reference point, my voltage reading for the same supply would be different and this would have consequences, right?


Yes, but if you're buying a power supply it's usually pretty clear which terminal is the reference terminal.