Currently, without any configuration, encrypted messages contain key ids of
senders and receipts which are visible to anyone.

I don't have a deep technical knowledge, but I would like to see gpg to
implement some kind of hashing rather than putting ids in plaintext. Is there a
way to do this, that each time a different hash will be created for the same ids
and it will be easy for the recipient's gpg app to see it's target private key
belongs to a private key in the keychain and simply ask password for that
specific key when decrypting.

If it's not possible to create different hashes for the same keys, even a fixed
hash would still be better.

This could be implemented as the default behavior, are there any drawbacks?