Are water boilers dangerous when showering?
I'm from Serbia, just like the OP, and we do have such a myth there. After my initial rant, aimed at explaining why some of the safety assumptions that many answers here may have are wrong, I'll show installation of a typical water heater and explain a couple of issues that I see with the installation.

(Feel free to skip this part)
First, some background, so that others can understand the issue:
Most water heaters are made by domestic or ex-Yugoslavian manufacturers. Furthermore, unlike the "western" countries, we do not have strict regulation of electricians and the "National Electrical Code", which so many Americans like to talk about, is not nearly as accessible here. The newest book on the topic that I could find was more than 20 years old. The result is that the regulations themselves are not easy to check for actual home-owners and there's a justified lack of trust in electricians, since a large number of them actually, and with no exaggeration at all, know less than someone who did a 10 minute Google search.

Next, there's the topic of electricians and superiors.
An electrical installation project for a building should be done and signed-off by a graduated electrical engineer that has a valid license as a "responsible designer of low and medium voltage electro-energetic installations".
The actual implementation should be supervised by an electrical engineer who has a valid license as a "responsible contractor of low and medium voltage electro-energetic installations".
I was unable to find if there's actual licensing for electricians that are doing the installation itself!  

Next we have a degraded civil engineering ecosystem: during communism there were huge companies making entire neighborhoods at one time, with many engineers and a strong internal structure. This changed and now we have small "investors" who are constructing one or two buildings at a time, focusing mostly on being as cheap as possible, with a huge flow of employees. The licensed EEs I talked about previously will sometimes just glance at the project done by someone else, sign it and take the money for that. The "responsible" contractors will never appear on the site to see what is done and the electricians are sometimes just some guys that know what to do with wires.  Safety inspectors are there mostly to take their share of the bribes as well. 

This was all assuming actual sort of legal building construction. In some cities, for many decades, it was pretty much impossible to get a construction license, resulting in entire settlements illegally built with no government oversight at all.
(End of background)

So the above resulted in low quality installations done quite often by uneducated people, or semi-responsibly done installations with heavily outdated safety standards. The majority of houses don't have ground fault interrupters and, disturbingly often, there's no ground in the bathroom at all!

Grounding systems are often TN-C, TN-C-S or TT, not uncommonly of the "rotten electrode" variety.

OK, so let's take a look at usual water boiler (as we call them here) installation:

Here's a "representative" photo of an actual installation from "Moja Radionica" website:
  

So there's a live and a neutral connection, if we're lucky a bi-metallic thermal switch (not pictured here), a regular thermostat, the heating element itself (sometimes several in parallel) and a neon indicator light. Today they also have a ground connection that's connected to the tank and outer chassis as well, but the pictured unit only has a ground connection to the heater. Usually the tank is also grounded through the pipes, which are often metallic, but the outer chassis in some designs doesn't have a good electrical connection to the tank and instead just sits on the glass wool, with the tank having wall attachment points.

The big hole on the picture is usually closed by a plastic hatch that attaches to the bottom part of the heater.

The heating element itself usually has resistance wire inside of a copper or nickel-plated copper tube that is connected to live and neutral via two connectors. The connectors are isolated from the tank itself with two ceramic insulators.

So in a properly functioning heater, there should not be any conductivity between the heating element connectors and the tank itself. One of the common issues is that the outer tube of the heating element cracks and water gets inside, often causing the resistance wire to break. This results in conductivity between the resistance wire and the tank itself, which could be potentially not very safe. Combine this with no ground fault interrupter or pipe-only or NO ground for the tank and then we have some cause for concern.

Another issue is the plastic hatch at the bottom, "neatness" of the wires and the rubber gasket between the heating element holder and the tank. Namely, these gaskets wear out and are sometimes not replaced after opening the tank, which results in leaks. The leaking water accumulates in the bottom of the heater and, depending on the neatness of the internal installation, may submerge some of the conductors inside of the water heater, which in my opinion can again raise some safety concerns. In a "Magnohrom" heater I've had, the wires were completely lying at the bottom of the heater and resting on the hatch. 

Another side-note about installation: This type of water heater should definitely not be installed somewhere out of the way, due to the safety valve. It's not shown in this picture, but on the cold water side, these heaters have a safety valve that is supposed to leak in the case of over pressure. Since many of the units do not have a thermal safety cut-off, in case the thermostat fails, this can cause flooding that might be undetected for some time. Furthermore, old valves do get stuck, so activating it by hand from time to time is recommended! If the heater is out of way, somewhere, this might be messy, this way discouraging the testing.