ECDH doesn't give you authenticity, i.e., you can't get a certificate for that.
When you look at ways to generate ECC keys, there are classes of vulnerable numbers for which you need to test.
When you look at ways to generate ECC keys, there are classes of vulnerable numbers for which you need to test.
ECDH doesn't, but ECDSA does. That's why we have Curve25519 & Ed25519, respectively.
I agree that any protocol that required you to generate curves or even base points on the fly would have similar concerns, but we generally don't use those kinds of protocols.
Still, while I'm punching here a wee bit above my weight - I should read the 186-4 again - there are some tests to verify the strength of the prime. Is it enough for a similar classification? - I don't know...
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.