What is the unit V/V?
Does it really mean, for example 10V per V (10V/V)?


Yes, it does. For every volt the input increases or decreases by, the output increases or decreases by 10 volts. Until it comes near the rails, of course. Note that this amplification is continuous; an input increase of 0.25V will result in an output increase of 2.5V, etc.