Within wine there is a huge array of glasses for different wines with the believe that different wines taste better out of different glasses.
This can also be seen (though to a lesser extent) with different beer mugs and even root beer (in the classic A&W mug).
How much does the flavor of coffee change based on the shape of the drinking
According to my "Lavazza Barista Guide" the shape of the cup affects the foam of the espresso and indirectly affects the taste.
It is explained as; the foam helps to protect the valuable ingredients to not to be evaporated easily. Also, the foam keeps you espresso warm while you are not drinking it.
In the guide, it is stated that the thickness and the vertical angle of the lavazza cups are specially designed for best coffee experience. So that, at each sip you drink some of the coffee and a bit of the foam and the remaining part of the coffee and foam fits to the now norrower part of the cup.
This sounds like a bit advertisement and a bit reality in a manual prepared by a commercial company. Quite expected in my opinion.
Since @MTSan already gave an excellent answer, I thought I would throw a twist into the discussion.
While you asked whether or not the shape of the drinking vessel would impact the coffee's flavor, this leaves out an important aspect of the cup you're drinking from: it's color!
While there are a few different sources out there, here's one research paper on the subject I found.
In it, the authors note that:
Both experiments demonstrate that the colour of the mug affects people’s ratings of a hot beverage...
And:
...it is possible that colour contrast between the mug and the coffee may have affected the perceived intensity/sweetness of the coffee."
While all such research should be taken with a statistically significant grain of salt, it's still food for thought.
I hope my slant adds to this Q&A! Comments and edits are encouraged.
