Ice cream at the food industry level is super cheap. It can be as cheap as 3 gallons for $15 or so, if you buy enough. I don't have an answer as to the availability of the AYCtE Shake at the Plaza, but it's not as expensive on their end as it seems.
The current "scoop" on the dining plan says you can order "specialty beverages" on the plan. I'm not sure if that AYCtE milkshake is a "specialty" beverage... the menu has it in the "cake, pie and shake" section, not the "beverages" section.
Can't say for sure, but last time we went to a restaurant on the deluxe dining plan - can't remember if it was 50s or sci-fi but definitely one of those 2 - the all you can drink milkshake was offered as the beverage.