Is it possible to have assigned different materials to different faces of an object, but to use only one single material as a bump map for the whole object? Thank you in advance for your answer.
You could use the exact same chain of nodes in all your materials. To make it more convenient you could group all these nodes in a node group (CtrlG to group, Tab to exit the group, CtrlAltG to ungroup), so that if you ever need to make a change it will be replicated in all the instances of this group in the other materials:
If you are already using UVs for your materials, you need to create a brand new one in the Object Data > UV Maps, that you'll use for the Bump node only: In the Shader Editor you just need to create an Input > UV Map, select the UV map you've just created, and plug it into the bump Image Texture:
The usual thing to do is to use a single material that would have different shaders mixed by some mask so you do not need to be limited by boundaries of faces on your model. You can define as many different shaders in one material as you need to.
There is however nothing keeping you from using the same texture in a few materials as well so you can also have different materials with the same bump map texture and that will be fine as well. Simply use the same texture in both of them with the same UV map or same texture coordinates of any kind and same settings.
Here is a material with two different shaders mixed with noise texture but sharing the same bump texture:
It doesn't need to be noise, it could be an image texture mapped with UVs instead.
If you made your question a bit more specific, you could get a more specific answer.