The resistance (R) of an object is defined as the ratio of the voltage across it (V) to the current that flows through it (I), so why can't I claim that the internal resistance of a voltage source is: r=V/I? (V is the voltage across the voltage source and I is the current that flows through the voltage source)
Because voltage sources are more complex than resistors and they don't follow Ohm's law.
In the case of a resistor, the resistance is causing the voltage drop. So measuring the voltage drop (along with other information) measures the resistance.
In the case of the voltage source, there is a resistance causing some voltage drop, but it is attached to some additional mechanism that is also changing the voltage. When you measure the voltage difference, you are measuring the sum of both mechanisms. To assign this sum to a single mechanism (the resistance drop) is incorrect.
$R = \frac VI$ is valid for ohmic materials in many (but not all) situations. A voltage source is not ohmic and you shouldn't expect Ohm's law to apply.