Is it safe to reuse ECDH asymmetric keys for authentication?
Such keys are called static keys. Keys that are newly generated each time are called ephemeral keys. Note that you need to trust the public keys of the static key pairs to use them for authentication. This trust can be established by embedding the DH public keys in leaf certificates within an PKI.
Please note that there is an issue when static keys are used on both sides during common Diffie-Hellman key agreement: the generated secret will be static as well, as the whole scheme has now become deterministic. The solution for this is to send a nonce as well as a public key by one of the parties; this nonce can then be used during key establishment to create a unique secret key.
More information in chapter 6 of NIST SP 800-56A, specifically 6.3
Scheme Using No Ephemeral Key Pairs, C(0, 2) scheme.