MonoGame is integrated inside Visual Studio.

To download and install MonoGame you need to click on the Extensions → Manage Extensions menu inside Visual Studio.

 
To download MonoGame you need to click on the Extensions → Manage Extensions menu inside Visual Studio.

 
1. Select Online
2. write monogame
3. and press Download
4. We need to close this window and close Visual Studio

 
After Visual Studio is closed, MonoGame will automatically start downloading:

 
Click on Modify to agree to the license.
And the installation of MonoGame will continue.

Press Close.
The installation of MonoGame is complete.
 

Answer: MonoGame is only visible when creating a new project inside Visual Studio.
 
Let's open Visual Studio:

Previously, the old MonoGame library was downloaded as a separate exe file and had to be run in order to install.
After installing MonoGame on the Windows system, it is not visible, can only be seen when creating a new project inside Visual Studio

On the official website MonoGame you can find any version:
http://www.monogame.net/downloads

I used version MonoGame 3.6 for VisualStudio http://www.monogame.net/releases/v3.6/MonoGameSetup.exe

Installing MonoGame 3.6 step by step:

Open the downloaded file and install MonoGame:

 
Press Finish
The installation was successful
Read: Create a new MonoGame application for Android...