Indeed, the expected number of fix points is the same for both permutations and mappings. Quite curious when you think about it, because the distribution of fix points is obviously different! (Consider the case n = 2 for the simplest example: there are mappings with exactly 1 fix point, but every permutation has either 0 or 2 fix points).
Note though that a correct block cipher is necessarily a permutation, because it's invertible (by definition, a permutation is just an invertible mapping with domain and range equal). A hash function on the other hand needn't be a permutation even when you restrict the domain to inputs of the same bit length as the hash output.
Note though that a correct block cipher is necessarily a permutation, because it's invertible (by definition, a permutation is just an invertible mapping with domain and range equal). A hash function on the other hand needn't be a permutation even when you restrict the domain to inputs of the same bit length as the hash output.