How to create behavior like the $HOME variable
You need to set a shell option for this
shopt -s direxpand

either directly on the command line or in your startup files.
From man bash
direxpand
   If set, bash replaces directory names with  the  results
   of  word  expansion when performing filename completion.
   This  changes  the  contents  of  the  readline  editing
   buffer.   If not set, bash attempts to preserve what the
   user typed.