Strictly speaking 'faire les courses' and 'faire les magasins' is to go shopping, the first one normally relating to food and essentials and the second one more for department stores and boutiques.
'Faire du shopping' is an addition to our vocabulary and to me means to go to the shops, not necessarily with the intention of buying. There is another expression meaning the same thing 'faire du lèche-vitrine' (to go window-shopping) which uses 'du' too. Maybe that's the reason?
Hope this helps!
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