AES Key Length vs Block Length
A block cipher is pretty much a substitution cipher. So let's look at a simple alphabetic substitution cipher. There are 26 different plaintexts and 26 different ciphertexts. The cipher is a permutation of these 26 values. But that does not mean there are 26 different permutations, it means that there are $26! \approx 2^{88.4}$ different permutations, which allows an 88-bit key without equivalent keys.

If you transfer this to an n bit block cipher, which has $2^n$ different inputs/outputs, then there are $(2^n)!$ different permutations, which is ridiculously large ($\approx 2^{2^{135}}$) for $n = 128$, much larger than the $2^{256}$ different permutations addressed by a 256 bit key.

So even a 256-bit key only uses a tiny fraction of the available permutations, and thus there are (probably) no equivalent keys if the cipher is ideal.