Ah! So
you weren't claiming to prove anything, just stating what you have to prove? But you called it your "solution", so I assumed you had defined what function you meant by 1, and proved that it is an identity.
Please give it some thought, and
tell us what you think the identity element might be! This is your task, not ours. And please ask specific questions so we can help you with your thinking.
You appear now to be asking
what the operation means. You gave the definition:
(f⊙g)(x) = f(x)⋅g(x)
This means that the "product" of any two functions f and g is the function f⊙g whose value for any input x is the product of the values of the two functions for that input, f(x) and g(x).
Take an example, with n=2. Then B is the set of all functions from {1, 2} to {0, 1}. We can describe such a function by merely listing the values of f(x) for x=1 and x=2; it can be written as an ordered pair, (f(1), f(2)). For example, we can write the function defined by g(1) = 0 and g(2) = 1 as g = (0,1).
Here is the set B: {(0,0), (0,1), (1,0), (1,1)}. Those are all the possible functions (ordered pairs). One of them will be the identity!
To multiply two functions, we just multiply corresponding elements.
The product of the functions f=(1,0) and g=(1,1) is (1⋅1, 0⋅1) = (1,0). Do you follow that? We have (f⊙g)(1)=f(1)⋅g(1)=1⋅1=1, and (f⊙g)(2)=f(2)⋅g(2)=0⋅1=0. In general, f⊙g = (a,b)⊙(c,d) = (ac,bd).
Now, what specific function g could you multiply by
any function f, and the result would be the same function, f? That g will be the identity.
By the way, what I am doing is showing you how to think about a problem like this. You first have to make sure you understand what the notation means, which you can commonly do by "playing" with the concepts, often using a small example like my n=2. This makes it more manageable. Once you get an idea of what the problem is about, you can start thinking about the specific questions that are asked.