Operational amplifier UA741CP - reading the datasheet
If you read your reference material's next paragraph:


  Some op amps, for instance, will only allow the common mode voltage of
  a signal to come within a diode drop or so of the power supply rails.
  


In your case, its more than a diode drop, if either V1 or V2 gets to within a few volts of the power supply rails, the op-amp goes non-linear. Those few volts of no-man's land are there by design and do not scale with supply voltage as you have suggested.
At op-amp output, the same situation exists. If output voltage swings up towards supply voltage, it needs a few volts to remain linear. The output cannot supply current to pull the output voltage higher. Again, this required headroom doesn't scale with supply voltage.
Internally, op-amps use entirely different circuit topology to achieve rail-to-rail performance. For something like a hearing-aid, where supply voltage is a 1.5v battery, op-amp design is quite an art.