Why didn't Minato's shroud change when he used Tailed Beast Mode? Naruto's Nine Tails Chakra Mode:


Naruto's Tailed Beast Mode:

 

The change in the appearance is very clear.

Minato



Minato's Nine Tails Chakra Mode and Tailed Beast Mode (when he can use Kurama avatar) appearance remains the same. Only his pupils change. So why didn't his shroud get a new design like Naruto's?
According to the wiki, the difference between "Nine-tails Chakra Mode" and "Kurama Mode", the one you called "Tailed Beast Mode" is simply that Naruto learned Kyuubi's name: Kurama. At that point, Kurama wasn't just lending Naruto chakra, they were fighting together.

I would guess that Minato never learned to fight together with Kurama or ALWAYS fought together with it, so Kurama never changed Minato's appearance.

The wiki actually defined "Tailed Beast Mode" when the Jinchuuriki assumes their tailed beast's form:



This is both Naruto and Minato in "Tailed Beast Mode" performing a combined Rasengan.