Can the Same Function Have Different Asymptotic?
The asymptotic behaviour of a function always depends on the parameter in the function. So for your example, I'd say that claiming that the function is exponential w.r.t. $k$, but polynomial w.r.t. to $n$ is correct.

When we look at the running time of algorithms, we consider it to be a function of the size of the input (usually the number of bits in a binary encoding). So, for an algorithm, it is always well defined whether it is for instance polynomial or not, but this can depend on how you define or encode your input.