https://en.wikipedia.org/wiki/Mille-feuille It doesn't cut nicely for fork and knife. When I tried eating it with my hands like a sandwich, the fillings of cream sploosh and the layers of wafer accordion. 
Fundamentally the answer is just 'it is added for flavour; you can skip it if you want'. A few suggestions as to why it might be a good decision to add it:  Vanilla flavour can be stronger or weaker, and it might be that the recipe writer prefers their milkshake to have a stronger vanilla flavour than the ice cream they buy. Depending on how much milk is being added, this will dilute the ice cream flavour a little, so more vanilla could be added to compensate. The flavour profile of the ice cream changes as it melts, because some flavours are subdued in a very cold product (in particular, the sweetness is reduced a little, which is why melted ice-cream or ice lollies are often surprisingly sweet; more sugar is needed so that the frozen product still tastes sweet). It might be that when the ice cream is no longer solid the balance is 'off', so more vanilla is needed.  