Transparent material like two-way mirror for specific colour
You could make something that works in the manner you describe, by placing monitor behind a device known on eBay as Electric Self-Adhesive PDLC Smart Film Glass. From the description, when power is applied, the film is transparent, while it is opaque/translucent when power is off.
I found that the product is almost exclusively white on eBay, with one entry for the product in black. A general search shows at least one manufacturer with an availability of nine colors.
The concept works with mirror film, because the level of light on the viewer side of the film is always greater than the level of light on the back side of the film, which is typically wall or some other panel. The exception to this that makes the mirror film work is that the monitor/display device generates its own illumination, providing the illumination difference sufficient to pass through the mirror film.
In the case of the PDLC smart film, there is not the requirement to have greater illumination levels, or more accurately, the requirement to have darkness behind the film. When powered, it will turn approximately transparent, presenting your monitor and anything else covered by the film.