What video card/driver version are you using?

Currently, I'm using 384.111-0ubuntu0.16.04.1 for x86_64, NVIDIA propietary driver with a NVIDIA GeForce 840M.

For mesa:
Did you also sudo apt update and sudo apt upgrade after sudo dpkg --add-architecture i386? After that, what does sudo apt install libgl1-mesa-dri:i386 libgl1-mesa-glx:i386 in a terminal give?

I used sudo dpkg --add-architecture i386 before, used it again and did sudo apt update and sudo apt upgrade. Nothing new, all is up to date. When running sudo apt install libgl1-mesa-dri:i386 libgl1-mesa-glx:i386 I get this:

Reading dependency list... Done
Making dependency tree
Reading status information... Done
Couldn't install some packages. This could mean you asked for an 
impossible situation or, if you are using an unstable distribution, some of 
the needed packages were not made yet or where taken from «Incoming».
This information could help to fix this situation:

These packages have breached dependencies:
 libgl1-mesa-dri:i386 : Rely on: libc6:i386 (>= 2.22) but it won't be installed
                        Rely on: libdrm-amdgpu1:i386 (>= 2.4.80) but it won't be installed
                        Rely on: libdrm-intel1:i386 (>= 2.4.38) but it won't be installed
                        Rely on: libdrm-nouveau2:i386 (>= 2.4.66) but it won't be installed
                        Rely on: libdrm-radeon1:i386 (>= 2.4.31) but it won't be installed
                        Rely on: libdrm2:i386 (>= 2.4.75) but it won't be installed
                        Rely on: libelf1:i386 (>= 0.142) but it won't be installed
                        Rely on: libexpat1:i386 (>= 2.0.1) but it won't be installed
                        Rely on: libgcc1:i386 (>= 1:3.4) but it won't be installed
                        Rely on: libglapi-mesa:i386 but it won't be installed
                        Rely on: libllvm5.0:i386 (>= 1:5.0~svn298832-1~) but it won't be installed
                        Rely on: libsensors4:i386 (>= 1:3.0.0) but it won't be installed
                        Rely on: libstdc++6:i386 (>= 5.2) but it won't be installed
                        Rely on: zlib1g:i386 (>= 1:1.1.4) but it won't be installed
 libgl1-mesa-glx:i386 : Rely on: libc6:i386 (>= 2.8) but it won't be installed
                        Rely on: libdrm2:i386 (>= 2.4.75) but it won't be installed
                        Rely on: libexpat1:i386 (>= 2.0.1) but it won't be installed
                        Rely on: libglapi-mesa:i386 (= 17.2.8-0ubuntu0~16.04.1) but it won't be installed
                        Rely on: libx11-6:i386 (>= 2:1.4.99.1) but it won't be installed
                        Rely on: libx11-xcb1:i386 but it won't be installed
                        Rely on: libxcb-dri2-0:i386 (>= 1.8) but it won't be installed
                        Rely on: libxcb-dri3-0:i386 but it won't be installed
                        Rely on: libxcb-glx0:i386 (>= 1.8) but it won't be installed
                        Rely on: libxcb-present0:i386 but it won't be installed
                        Rely on: libxcb-sync1:i386 but it won't be installed
                        Rely on: libxcb1:i386 (>= 1.9.2) but it won't be installed
                        Rely on: libxdamage1:i386 (>= 1:1.1) but it won't be installed
                        Rely on: libxext6:i386 but it won't be installed
                        Rely on: libxfixes3:i386 but it won't be installed
                        Rely on: libxshmfence1:i386 but it won't be installed
                        Rely on: libxxf86vm1:i386 but it won't be installed
E: Couldn't fix these problems, you held broken packages.

And now what? I guess the issue you referenced here is not a Steam problem, so I just should wait or ask elsewhere...

Cannot get what are you talking about... There are a lot of people like me who cannot handle this problem, but I dunno what to do... >.<

Run this in your terminal;
sudo apt install mesa-demos:i386
glxinfo | grep OpenGL

It should list some info about your GPU if it succeeded

How did you install steam? I recommend installing it with apt if that isn't what you already did.

@Criminalllz
Trying to install mesa-demos:i386 prints this:

Reading dependency list... Done
Making dependency tree
Reading status information... Done
E: It was not possible to find the package mesa-demos:i386

How did you install steam? I recommend installing it with apt if that isn't what you already did.

Installed it by .deb file in Steam webpage and by apt. All gives the same result...

mesa-utils:i386*

My bad.

@Criminalllz so sad :(

Reading dependency list... Done
Making dependency tree
Reading status information... Done
Couldn't install some packages. This could mean you asked for an 
impossible situation or, if you are using an unstable distribution, some of 
the needed packages were not made yet or where taken from «Incoming».
This information could help to fix this situation:

These packages have breached dependencies:
 mesa-utils:i386 : Rely on: libc6:i386 (>= 2.4) but it won't be installed
                   Rely on: libgl1-mesa-glx:i386 but it won't be installed or
                            libgl1:i386
                   Rely on: libx11-6:i386 but it won't be installed
E: Couldn't fix these problems, you held broken packages.

Almost all packages that Steam asks me to install when I run it, rely on packages of 32bit architecture. These packages are installed but in their 64bit version, which came installed I guess in the OS, as my OS is 64bit...
The thing is the 32bit package has a problem with the 64bit one, so one dependency is not completed, neither are the rest...

What happens if you use aptitude in place of apt?

It's just ok. Asked outside and someone told me to install Steam alongside Flatpack, and now it works. Thank you anyways :)

I think you will need to install other 32-bit things somewhere down the line, so i would recommend attempting to fix the problem properly.