Superdense coding - What are the individual qubits A and B?
Is it just that A and B cannot be written using the standard basis?


Correct.
Entangled qubits are not separable.
You can't factor them into a form like $(a_0 |0\rangle + a_1 |1\rangle) \otimes (b_0 |0\rangle + b_1 |1\rangle)$.

You can get a per-qubit density matrix by tracing out the other qubit, but the resulting density matrix doesn't capture the correlation between the two qubits. It just looks like a maximally mixed state.