Bathroom mirror on wall outlet

Greetings!

I feel like this is an absolute beginner question, but: Is it possible to connect illuminated bathroom mirrors to an existing wall outlet so that I can control the bathroom mirror with the bathroom light switch?

I have a wall outlet here (like a ceiling outlet for lighting, with N/L wires and grounding). However, the actually intended bathroom mirror came with a power plug for connection to a socket. That way I would only have been able to switch the lighting on the mirror itself. That was not what I was looking for.

Unfortunately, I rarely find hints online about connecting the respective illuminated mirror and I am generally wondering whether the wall outlet in this form is actually only intended for a separate lamp or whether I can/must somehow "modify" the 3-wire wall outlet to achieve my goal.

In the past, I have indeed always had mirrors without lighting...

If the existing wall outlet is switchable, no problem. Unplug and connect to the wall outlet. Done.

If the existing wall outlet is switchable, no problem.
Unplug and connect to the wall outlet.
Done.

not quite finished - in damp rooms the connections must be absolutely securely insulated with heat shrink tubing to prevent problems caused by leakage currents. This is not as simple as, for example, living room lamps.

not quite finished - in damp rooms, the connections must be absolutely securely insulated with heat shrink tubing to prevent problems caused by leakage current. This is not as simple as, for example, living room lamps.

I would like to know the source for this rule... What describes here is probably protection zone 3, so the only thing that really has to be installed is an RCD.

Source please! "I heard once" is not a source :)

What describes here is probably protection area 3, so the only thing that really has to be installed is an RCD.

Personal experience. A burnt child in the best sense of the word. In the last rented apartment, the renovation was carried out by a trained baker. He connected the bathroom cabinet with lighting exactly as described above. He also did not install a moisture-proof lamp for the ceiling light. When you touched the wet tiles after showering, you got an electric shock.
The cabinet was then connected by a specialist company.

What describes here is probably protection area 3, so the only thing that really has to be installed is an RCD.

Isn't the RCD mandatory in the bathroom anyway? So it should already be there.