How to select the whole variable name including $ in Visual Studio Code in PHP?

The best way is to edit the VsCode settings.json and specify it for the PHP language.
You can open it typing VsCode command:

Preferences: open settings (JSON)

And specify inside the “editor.wordSeparators” setting for your language removing the ‘$’ symbol:

"[php]": {
  "editor.wordSeparators": "`~!@#%^&*()-=+[{]}\\|;:'\",.<>/?"
}