Enlighten Me Please Regarding PGP Encryption

I have somebody’s public PGP key imported on my laptop (Debian Linux), but not their private key. When I encrypt a file to them using their public key, I can then decrypt it again even though I don’t have their private key. Through experimentation, it appears that my secret key also works to decrypt the file, even though I selected the other person as the recipient. Is this intended behavior? How can two different private keys decrypt the same file? Is it actually using AES encryption to encrypt the data and then just locking the AES keys behind our respective PGP keys? This would also explain why, in PGP encrypted chat apps like XMPP, I can read my own messages on devices that I didn’t send it from without compromising the security of the message, but my basic understanding of asymmetric encryption was that once something was encrypted with somebody else’s public key, nothing but the intended persons’ private key could decrypt it.