No, this isn't an oversight. AES is a block cipher, which is a keyed permutation. Now if you have a permutation of, say, three elements there are e few permutations possible:
a -> a
b -> b
c -> c
but also:
a -> b
b -> c
c -> a
and
a -> c
b -> a
c -> b
and
a -> c
b -> b
c -> a
(there should be $6$ for $3!$, the factorial of three, I won't write them all out)
Now if you look closely, there are more than 3 permutations possible. The actual number of permutations is higher than the input/output! To be precise, the number of permutations is $(2^{128})!$, which is a lot more than $2^{256}$.
The purpose of the key is to select one of the possible permutations. So the key could be a lot larger than 256 bits and still be meaningful. This also means that there may be multiple keys that map a specific ciphertext block to a specific plaintext block. An attacker may have to validate more pairs to be reasonably sure of success. For instance, in both the last two permutations in the example, a maps to c.
The block size is - as Yehuda Lindell writes in the other answer - needed when many blocks are encrypted. For instance when you choose a random nonce for CTR mode, the blocksize must be high enough or you will get overlaps in the counter that is encrypted to create the key stream (i.e. choosing a twice in the example). In that case the key stream would repeat, breaking the stream cipher.
An example of a block cipher with a much larger maximum key than block size is Blowfish, which has a block size of 64 bits and a key size of max 448 bits. Blowfish is considered deprecated by the author, Bruce Schneier. Besides other issues, the small block size is a major reason for the deprecation. Both followup ciphers, Twofish and Threefish, expand the blocksize.
