The human body has a very non-linear voltage/current response. Ohm’s law only applies to devices with a (mostly) linear response.
A very simple model of the human body is a resistor in series with a resistor and a capacitor. The resistor kinda simulates the skin and the rest simulates your internals (it’s very much an over-simplification). However, the value of these resistors varies depending on the applied voltage.
At a low voltage like USB voltages (5 volts) the human body has a very high impedance. It varies a bit from person to person, but generally we’re talking several hundred thousand ohms to several million ohms. If you take your handy dandy multimeter (which also uses a very low voltage) and measure the resistance from one hand to another, you’ll get a very high value. If you use this value to calculate how much current you’ll get from a 120 volt line though, you’ll be way off. At 120 volts, the human body’s impedance drops down to about 1,000 ohms, which again varies a bit from person to person and the conditions.
Generally speaking, voltages start getting dangerous to touch above about 50 volts. You can grab both terminals of a 12 volt car battery and not get shocked. 5 volts from a USB cable is pretty darn harmless, and (if the USB port complies to the spec) the current should be limited to 500 mA or so anyway, which further prevents possible damage. Even so, there is a small risk of fire. 500 mA through a small enough piece of metal in a bad connection can get quite hot.
As others have said, there’s also a risk of damage to the equipment or the port.
By the way, the official “safe” current as defined by most safety standards is 5 mA. A 10 mA current isn’t likely to cause your heart to go into fibrillation, but it’s not guaranteed not to either.
If the insulation is damaged but the wire underneath is fine, simply covering the damaged insulation with electrical tape is sufficient to repair it. If the wire underneath has been damaged, it needs to be repaired or replaced.
ETA: A heat gun and some heat shrink tubing, as cmyk said, works great to repair insulation damage as well.