No, this is a misleading oversimplification. Humans are high resistance. Current can't flow through a person without a high voltage to drive it.
It's not the voltage or the current that kills you; it's the energy.
Yet when you scuff your feet across a carpet on a dry day, you are charging yourself up to thousands of volts. When you then touch a grounded metal object, the discharge can send several amps of current through your body:
about 5 to 7 kilovolts was the maximum value measured on the human subjects. ... asked him to shuffle his feet while connected to the electrostatic voltmeter. Much to the EMC lab’s surprise, the voltmeter registered 18000 volts! A Brief History of Electrostatic Discharge Testing of Electronic Products
It should be mentioned that the reference model of the ESD waveform is the human-metal discharge. ... The maximum ESD current value is 12 A, whereas the IEC Standard defines 15 A. Electrostatic Discharge Current Linear Approach and Circuit
This is far more than 25 V and 70 mA, so if those can each kill you, why doesn't ESD? Because the duration of the discharge is a fraction of a microsecond, and the total energy release is only a few millijoules. It doesn't have enough time to cause fibrillation or to heat up and burn significant amounts of tissue.
"The effects of electrical current passing through the human body are covered at length in the International Electro Technical Commission document IEC 479-2:1987. In this document it indicates that a transient or capacitive discharge, as is the case with static electricity, requires energy in excess of 5 Joules (5000mJ) to produce a direct serious risk to health." — Static electricity in modern buildings
The reason this "safety tip" is terrible is that it misleads people into thinking that high-current power sources are dangerous to touch and high-voltage power sources are not.
Most power sources are voltage sources, not current sources. This means they output a constant voltage, and the current in the circuit depends on the resistance of the load (the human body, in this case). This is true for power lines, batteries, etc. Most people don't understand that the current listed on a power supply is just a maximum rating, and won't actually go through their body if they touch it.
If you connect a 1 kΩ resistor across a 12 V supply, the same 12 mA of current will flow whether the supply says 100 mA or 100 A. The amperage rating of a power supply just tells the current it could source, if connected to a small enough resistance. It doesn't force that amount of current through anything it touches, or it would be constantly arcing through the air.
Yes, car batteries can source a lot of current (hundreds of amps), but this only happens when they're connected across a small resistance. If you connect a screwdriver across the terminals, a huge current will flow and the screwdriver will melt, battery will explode, etc. If you put your hands across the terminals, nothing will happen. This is because your skin's resistance is much higher than the screwdriver's resistance. So the 12 V 600 A car battery will not harm you because the voltage is not high enough.