The prime field is part of the definition of the curve. Everyone interoperating on that curve shares it. It may have been generated weirdly --- the Curve25519 prime sure was! --- but all that tells you is what curve they're using.
It's true that the particular curves a system supports could be a kind of fingerprint, but that's a banal observation, equally true of the ciphers they support, or the compression algorithms.
It's true that the particular curves a system supports could be a kind of fingerprint, but that's a banal observation, equally true of the ciphers they support, or the compression algorithms.