After creating an account (i.e. lets say using Mist) we can update the account name from the Mist interface by editing it. Is there a way that we can fetch that account name by providing the account address Web3 API. Hoping this is different to ENS.
No, the name that you assign to an account is stored in a config file used by Mist. Web3 only interacts with the backend node that Mist connects to, so it doesn't have any access to things like wallet names.
This is not the same as ENS, which you would be able to interact with via Web3, but is an entirely different system (sort of like the difference between the name you assign to a bookmark in your browser vs. the name
google.com used by everyone to actually look up the (ip) address.)