I wonder what are the practical differences between using Authenticated Encryption and then ECIES. It looks like both require communicating the shared secret in some fashion between the parties and both provide encryption and authentication.
ECIES is a specific hybrid encryption scheme using elliptic curve cryptography to enable EC based encryption - that is: encryption using the static public key of the receiver and decryption using the static private key of the receiver.
AEAD is a description of an interface for symmetric encryption with authentication.
ECIES depends on a asymmetric key pair (where the public key is trusted) and AEAD depends on a single symmetric key, which needs to remain secret. ECIES is a specific construct to perform encryption using Elliptic curves, AEAD is a description of an interface for authenticated ciphers. In such, it doesn't make much sense to compare the two.
To make a concrete example, consider we have a party A, which receives messages {x1, x2, ..., xn} from various other parties, but needs to read the sender from the message itself, because upon receiving a message x A doesn't know the sender. Using some AEAD scheme, A could have a plain-text part in each x telling the sender in an authenticated part and hence would need which shared secret to choose to open the actual message.
You wouldn't really need to authenticate the key identifier; if you'd change the key identifier then the authentication tag is already invalid, as you would use the wrong key to authenticate the ciphertext. You'd only need it if the scheme reuses the key for multiple senders that trust each other (not a common scenario). If this is for transport authentication you'd generally just keep track of a session.
Can ECIES do this and if so, what is the practical difference? If not, well, what could be the purpose of ECIES? There seem to dearth of information on ECIES.
No, ECIES just performs encryption, and as anybody has the public key, anybody can create a valid ciphertext. It doesn't If you want to add such functionality, you should sign the data, and then encrypt the plaintext and signature.
If you're talking about real time communication between a sender and receiver then you'd just use a transport protocol such as TLS. TLS 1.3 requires AEAD but since there already is a key establishment scheme based on DH, ECIES isn't used.
