Did we ever actually prove that every Turing machine can decide/accept only one language?
It's almost axiomatic.
The language of any Turing machine is defined as the set of all strings (over the given alphabet) that the machine accepts (whether it be a machine that always decides or not).
That is a well defined set. There is no other set of such a description.
But if you want something "more formal" then assume two languages fulfil the above definition: L1 and L2, such that they are different. I.e. there is a string w in L1 (without loss of generality) and not in L2.
By definition the Turing machine doesn't accept w because w isn't in L2 (again, unimportant whether or not the machine halts on this string, since we're not simulating it, we're discussing the property that is empirically either true or false).
However w is in L1, so the machine does accept w.
This is a clear contradiction.
We see that the original assumption that 2 such sets fulfilling the definition that are different is wrong, and conclude that any 2 such sets must be equal, and thus the language of any given Turing machine is unique. I.e. There is only one.