ECDSA Public key randomness
ECC public keys are not random numbers.

They are generated through a scalar multiplication of a random scalar with a known/public point on the curve, called Generator.

The entropy lies entirely in the random scalar. Two public keys will collide if the random scalars collide.

When you say "256bit ECDSA  public key" you probably mean that your elliptic curve curve is defined over a finite field whose size is approximatively $2^{256}$. This would explain why you are mentioning a public key of 64 bytes in your question.

Now, in cryptographic secure elliptic curves, that size is closely related to the size of order of the generator, which is actually the size of your random scalar.

Therefore, without knowing the exact curve you are using I can't exactly answer the question. But approximatively the expected probability is: $\frac1{2^{256}}$

This has nothing to do with the encoding, so xoring the key over itself doesn't really make sense to me, or I didn't understand it.