Towel rail

It could be voltage induced in the towel bar surface (electricians say phantom voltage). But a finite number of milliamperes will flow if you "short" the metal towel bar to the nearby metal faucet. It is not unusual to feel a power line frequency vibration when rubbing your hand on the bar while your bare feet are flat on the floor.

High impedance voltmeters (most digital models) are very apt to show a noticeable voltage reading from an object to ground from induced voltage.

Now if the number of ma that would flow if you shorted the bar to the tap is high, say, more then ten, we have excessive leakage, enough to count as a genuine fault from a live part inside to the metal bar. That is a defect.

Connect a small wattage, say, 7 watt, incandescent lamp between the bar and the tap. If it glows, not necessarily brightly, or trips a ground fault current interrupter then the current leakage is great enough to be a fault.

If the lamp does not glow then you can use the in-line amperes function of a multimeter to measure the number of milliamperes. The subject subcircuit is between the hot feed to the towel bar and the faucet and the load is the dielectric (insulating material between the energized part(s) inside and the exposed bar.

When you do just the meter measurement, you create a subcircuit with two loads in series: the dielectric, and the meter itself. The larger the impedance of the meter compared with the impedance of the dielectric (at power line frequency), the larger the voltage reading on the meter, up to the supply voltage.

(If you did not do the lamp test and there was a real fault, the resulting current could blow out the meter during an in line amperes test.)

More than about 5 ma should trip a GFCI (RCD).

Installing the RCD (GFCI) removes the worry. This protects against electrocution since if the leakage is too great, the difference between outgoing and return current via the mains trips the RCD. Smaller induced currents will still flow if a circuit to ground, such as from your hand touching the bar to your bare foot on a not perfectly dry floor, is completed.

Yes, if the power plug (cord cap) is reversed in the receptacle (wall plug) you could sometimes feel more of a vibration under some circumstances. When the largest live parts inside or those live parts closest to the external parts are conneccted to the neutral side of the line (mains) then their voltage relative to ground is held close to zero and any induced current when you touch the bar will be less. The switch on the appliance properly cuts the "hot" side and when in the off position prevents voltage from being applied to the components inside and then no voltage is induced in nearby exposed parts. Most equipment in the U.S. has "polarized" plugs; one prong (the neutral) is larger than the other and the receptacle slots are sized to match. This prevents reversing the plug.

Is the circuit you hooked up the towel bar to grounded (has a ground wire aka equipment grounding conductor accompanying the circuit conductors to the breaker panel)? The ground wire should be connected to the frame (the bar itself).