Can elliptic curve cryptography encrypt with public key and decrypt with private key like RSA? I know that RSA can be used for both, encryption and signature.
What about EC? I know about ECDSA/EdDSA, but to my knowledge it can only be used to sign. 
I also know about ECDH, but it is a key agreement protocol.
Is there some elliptic curve based public key encryption algorithm?
Yes, there is a elliptic curve based public key encryption 
Let $a$ be Alice's private key and $a=aG$ be her public key. Bob, who wants to send an encrypted message to Alice, does the following :

Bob chooses a random number $r$, $1\leq r\leq n-1$ and computes $rG$
Bob then computes $M+r\alpha$. Here the message $M$ (a binary string) has been represented as a point in $\langle G\rangle$
Bob sends the encrypted text pair $\langle rG,M+r\alpha\rangle$ to Alice

On receiving this encrypted text Alice decrypts in the following manner

Alice extracts $rG$ and computes $a\cdot(rG)=r\cdot(aG)=r\alpha$
Alice extracts the second part of the pair $M+r\alpha$ and subtracts out $r\alpha$ to obtain $M+r\alpha-r\alpha=M$

There is a drawback in this as block of plaintext has to be converted to a point before being encrypted, denoted by $M$ above. After the decryption it has to be re-converted to plain text.