What version of Android starts “Messages for Web” support? The Messages for Web documentation gives the impression (as does the Chrome OS documentation on the matter) that this feature has been available since Android 1.0 as it lists no "if you have Android greater than". But my LG V20 is running Android 7.0 and in the Messages app under the "More icon" there is no menu item "Messages for web".
The "Messages for web" feature works only with the Messages app created by Google. 

The LG V20 runs a custom version of Android with a customized user interface. It is not the stock version of Android using only standard Google apps as is shipped on, for example, Google Pixel devices. The manufacturer (LG) will customize and add more features to applications like dialer, messages, camera, themes etc. creating a custom Android version above what Google provides. So, in this case, the Google SMS messaging application has a feature "Messages for web" which has not been implemented in the Messaging app created by LG.

But, you can use this feature by following these steps:


Download the Messages app by Google from the Play store.
Set the Messages app as the default messaging app in Settings.  
Open Messages app, and tap on three dots on top-right corner (options menu), select 'Messages for web'.
QR code window will open, enable camera permission, scan the QR code on the desktop, your messages on android device will be opened on your desktop/laptop.


Note: You can't uninstall the LG provided Messages app as it was installed as a system app.