Your code is doing as you expect but you are not making use of the loaded addin before quitting Word. If Word is subsequently re-opened, you will discover the addin is now in your list of addins but its status is not turned on.
You can either choose to turn this on every time you open Word by using just your .Installed line but if you want it ALWAYS turned on, it is easier to simply copy the template to the Addins folder.
By default the Addins folder is in %appdata%\Microsoft\Addins OR %appdata%\Microsoft\Word\STARTUP but it could have been changed by the user so this following code would perform the task of installing the template in a location so that whenever Word is opened, the addin is installed and active.
Code:
Set oWord = CreateObject("Word.Application")
Dim sAddins As String
sAddins = oWord.Options.DefaultFilePath(8) 'where 8 = wdStartupPath
FileCopy "C:\Program Files (x86)\Labelling Tool v.2.0\LabelingData\Labeling.dotm"", sAddins & "\Labeling.dotm"
oWord.Quit
Set oWord = Nothing