I design profile picture and cover photo for Twitter and I want to pick a color that I will want to use also on other platforms, such as Facebook and any other web page, but I want to be sure it will look the same on computer screen, smartphone and maximum close on printing materials. For example, I pick a color from pantone gallery and there is rgb and cmyk codes, is that safe to use these? And can I start a new illustrator document with rgb mode, fill object with the color from rgb code and then paste in cmyk document the same object and fix cmyk numbers, is that ok?
I don't actually see a question quite this broad though as Cai mentioned in comment there are a number of questions the cover various reasons on this in more specific use cases, color-reproduction
What it boils down to though is that color is based on human perception. People don't see colors the same, monitors don't see colors the same either. What color wall your in front of, the color and power of any lights around you including the sun, if your eyes are fatigued, if you even have perfect color vision to begin with (color blindness) --- all of these influence what color you perceive.
So even if you select a color within the most common parts of the RGB/CMYK color space and you'd still get variances. Combine that with 99.9% of people never calibrating their monitors and a clear 100% never calibrating mobile devices and any concept of accurate color consistency goes away.
Pantone will help get consistent colors on products. For example the Coke Red uses pantone so their shirts, vehicles, vending machines, signage, all of it gets the exact same red because pantone is carefully calibrated and essentially premixed to Coke's specifications. Even Coke though online or in advertising ends up with some level of variance. A Coke ad in a newspaper for example will be forced to use CMYK and might look darker and less vibrant than their Pantone would've.