No, they’re completely unrelated.
The sole developer of Shader Graph, Freya Holmer, effectively abandoned it due to being too busy making an actual game (Budget Cuts) and is not an employee of Unity. Unity did try to buy Shader Graph and hire her to continue working on it, but she declined as it wasn’t what she wanted to work on.
Funny enough, Shader Graph was not the first node based shader editor Unity tried to buy or developer they’ve hired. The very first node based shader editor for Unity was Strumpy Shader Editor, and they hired the developer of that, Tim Cooper, to make an official one sometime back in 2011-2012. However he never actually ended up working on that, nor did he work on the new official Shader Graph.
Shader Graph is written mostly by a single guy Unity hired just 2 years ago, Peter Bay Bastian, along with a small team of others who have made contributions.
Shader Forge is great, been out for along time, also now free, but is effectively abandoned and the main branch doesn’t work in Unity 2018, though others have tried to fix it to keep it working in forks of the project.
Shader Graph is new, and missing a lot of tools and features, but is also free. It also only works with the new LWRP and HDRP rendering pipelines and not the built in forward or deferred paths. Unity will hopefully continue to work on it, but it is pretty bare bones right now.
Amplify Shader Editor is great, been out for a few years now and has a ton of features. It works with the built in rendering paths as well as the new SRPs. It’s in active development with new features being added constantly has far more features than Unity’s Shader Graph so far, or even Shader Forge. It also appears to have a relatively active community that contributes new stuff all of the time on their forums.
My preferred shader editor is Sublime Text … because I like working with shader code directly and I’m crazy and don’t like IDEs.