Ed25519 or more general the EdDSA (Edwards-curve Digital Signature Algorithm) approach can be considered as a variant of ElGamal signatures (such as Schnorr or DSA). They all are signatures following the hash-then-sign approach. This simply means that you can sign arbitrary length messages by hashing them to a constant size string using a secure cryptographic hash function and then applying the signing algorithm to the hash value. Such as other ElGamal type signature schemes like ECDSA, when implemented using elliptic curve groups, the security relies on the elliptic curve discrete logarithm problem on the respective curve group.
The hash-then-sign approach, however, must not be confused with quantum resistant hash-based signatures such as Merkle signatures, Lamport signatures or the Winternitz one-time signature scheme. Such signatures do not rely on any (number theoretic) computational hardness assumption such as factoring or discrete logarithm assumption and thus are not susceptible to quantum attacks.