How much electricity is there in the "power switch" button cables?

The power switch provides an active low input to a logic gate within the power supply circuit.

This provides a low current logic low (ground) gate pulse to a BJT transistor biased to an isolated 5V source with the maximum current likely much less than 10mA. This portion of the circuit is also electrically isolated from the rest of the power supply circuit as can be seen in the diagram.

So to answer your question, No there are no dangerous voltages present.

Source: https://en.wikipedia.org/wiki/Power_supply_unit_(computer)

The wires should only have low voltages on them (<= 12V), so they're safe to touch.

However, you should be careful, because if one of the wires is connected to the standby voltage rail, shorting it to the chassis could potentially damage the motherboard.

The wires are safe to touch as they are powered from a 3.3V chip on the motherboard, and current is limited by a resistor.

However, if you carry an electrostatic charge, and you touch the wire, you will cause an electrostatic discharge (ESD). This is harmless for you, but it can zap the integrated circuits at the other end of the wire if they are not protected.

Normally, the manufacturer would have taken appropriate measures to ensure all input/outputs are protected against ESD, which begs the question: did they consider it worthy to protect this particular IO since it is normally not an externally accessible IO, but rather an internal wiring?

For example, looking at this motherboard schematic (pages 23-24), the power button signal is filtered by a 100nF cap which should be fine, however the RESET button signal goes straight to a IC pin without any protection.

While these chips include ESD protection diodes, such diodes are sized to protect against ESD events during fabrication, which are quite limited in charge as people manipulating chips are expected to take proper precautions. More robust protection diodes on every pin would use more silicon area and make the ICs more expensive, while providing no benefit unless the pin is actually accessible on the finished product and can be zapped by the user.

Therefore, your arrangement carries a risk of zapping the motherboard. It will probbaly be safe if the floor is tile and you are wearing cotton. However, if the floor is insulating, and you're wearing wool and sneakers, and the weather is dry, when touching metallic objects your fingertip can turn into a pretty impressive lightning gun!