So, I get this message in Photoshop when I try to insert values above 256 in RGB color picker menu, although I switched to 16 bits (image -> mode -> 16 bits per chanel). Why is that? Isn't 16 bits supposed to give me 65.536 colors per channel in theory?
The internally used values are 16 bit. The GUI uses only the 8 bit values, though. You can see that also in levels, curves, etc.
Still, when you enter 255, in reality this gets converted to 65535 (or at least some value >65280, i don't know their exact algorithm).
Yes, this means that only 256 of those 64k colours are directly numerically accessible for you. You can complain at Adobe...
The RGB colors are always 0-255 per channel... which is over 16 million colors and more than a human can discern. What those values mean depends on the color space... i.e. 0,255,25 in sRGB is the same color as 138,236,80 in ProPhoto.
16bit gives the potential for 65,563 tonal values... which is rather useless. The main benefit of using 16bit (or even 32bit floating point) is greater mathematical accuracy.
Edit to add: You can set the info panel to read out the colors in 16bit by clicking on the RGB eye dropper in that panel. But that's the only place where it will tell you what is really going on.