Are smoothies and milk shakes the same things with different names?
For smoothies, the main body of the drink is fruit that has been blended with some limited quantity of liquid or ice. Typically, one of the fruits is banana, and other fruits (berries, strawberries, kiwi, etc.) are added for flavor. They often include a sweetener of some sort, such as honey or turbinado sugar.

Milk Shakes are typically composed primarily of milk or ice cream, and may include limited amounts of fruit for flavor. They also usually include some sort of sweetener, such as sugar syrup or malt syrup, or a flavored syrup such as strawberry or chocolate. 

The general principle is that smoothies are mostly fruit, milk shakes are mostly milk. For your drink, I think it being a smoothie or a milk shake depends on the proportions of the ingredients added (i.e. more banana means smoothie, more milk means milk shake). 

Based on the general principles from above, you could call your drink either a milk shake or a smoothie and be correct. That said, the original Banana Smoothie, as recorded from 1941 on this smoothie fact page, included two big ripe bananas and two and one-half cups of milk. Since that seems to be exactly what you're making, I'd say that your drink is a smoothie based on the original definition of the term.