My stock shifter moved like yours until i installed an mgw shifter, have you seen the stock rear mounting location of the stock shifter? Its a cheap piece of steel which has soft rubber in it and the rubber is held to it with 2 plastic zip ties. I'll say it again, ford chose noise reduction over the quality shifts you would get with firmer parts in the areas of " motor mounts" "tranny mounts" and "rear shifter mounts". Thankfully the after market has fixes for all these issues that plague todays mustang. Trust me i have replaced my shifter with an mgw, and my motor mounts, and those 2 things have made a big difference in my shifting quality.
2016 Ruby Red GT/CS 401a automatic with 3:55 gears, bone stock, but not for long!