Yes, it is two factor authentication with a static second factor that will not be considered private by most users. And yes, a 'real' two-factor authentication mechanism would provide better security.

Unfortunately, due to market competition many websites simply cannot require 'real' two-factor authentication for all users. Here are the steps I would need to provide to my father to register for a typical '30-day free trial':

  1) Go to website.com and click 'Register'
  2) Enter your email address
  3) Think of a password and type it 
  4) Click 'I agree'
  5) Click 'Register'

Here are the steps I would need to provide to my father to register on a website for a free trial with 2-factor authentication using the Google Authenticator app:

  1) Go to website.com and click 'Register'
  2) Enter your email address  
  3) Think of a password and type it 
  4) Click 'I agree'
  5) On your phone, press the 'Play Store' or 'App Store' icon
  6) Press the 'Search' icon and search for 'Google Authenticator'
  7) Press 'Install' and wait for it to install (if you have an iPhone the install button might look like a little cloud icon)
  8) Press 'Open' to open Google Authenticator
  9) Press the 'Menu' button which looks like three dots in the top-right corner of the phone screen
  10) Choose 'Scan with barcode'
  11) Point the phone at the computer screen as though you were going to take a photo of the barcode on screen. 
  12) Wait for the phone to register the barcode, then enter the number shown on your phone into the website form
  13) Click 'Register'

Even with all these steps laid out for him, my father would probably find it extremely frustrating to get to step 13.