How to copy exact same black, grey and whites from a picture and apply these to a color picture?

Could you please set the Status Bar to »Document Profile« and post meaningful screenshots with the pertinent Panels (Toolbar, Layers, Options Bar, …) visible? 

Is the »grayscale« image actually grayscale or is it a desaturated RGB? 

What are the involved images’ exact Color Spaces? 

I've read this four times and I'm still not clear on what you want... 😉

Basically, you can layer them and use either "Color" or "Luminosity" blend mode on the top layer.

However. There's a caveat here. You must make sure all your images have an icc profile embedded, either an RGB profile or a grayscale profile. And in the latter case there's a gotcha.

Grayscale is subject to standard color management just like RGB. Obviously there's no color involved, but each grayscale profile has a specific tone curve (gamma). This will determine which numbers are produced from the visual tone, and this in turn again affects how it will interact when applied to other images. I would recommend Gray Gamma 2.2 if your color images are Adobe RGB, and sGray if your color images are sRGB or P3

But we're not through yet. Desaturating an RGB image just averages out the RGB values. It does not take into consideration that some colors are inherently bright, like yellow, and some are inherently dark, like purple. The result is that yellows turn into a drab and dark gray, and purples appear washed out.

The Lab color model works differently. It takes this into account, by separating the brightness component from the color components. So to get the same visual appearance in terms of tones, convert to Lab first and then desaturate. Then you can convert back to RGB. It's more easy to illustrate than to explain: