If I go to bitaddress.org and create a paper wallet with BIP38 and a pass phrase, what it doing with that pass phrase specifically? Since it can make multiple paper wallets from one pass phrase, the phrase must not uniquely identify the public and private keys. But if it does not, what use is the phrase?
Key pairs created from a passphrase is part of BIP39 and BIP32, which is completely unrelated to BIP38.
The word "Passphrase" for bip38 encryption can be misleading terminology because most people think of the 12-word mnemonic when they hear "passphrase" which is associated with bip32 HD wallets.
The encryption password you enter into bitaddress simply encrypts your one single private key so that it can not be used unless you decrypt it by typing in the password.
Since it can make multiple paper wallets from one pass phrase
The passphrase is not used in that way. Bitaddress can/will generate multiple paper wallets for you but the keys are not derived from the passphrase. If you elect to encrypt your private keys with a bip38 encryption password each key will be encrypted with that password. It is not creating any keys from the passphrase.
BIP038 is a encryption standard for Bitcoin Wallets, your passphrase is for deriving keys for that encryption. The private key generation works by other means, and is strongly not recommended to generate your keys in sites like bitaddress. Download some wallet like Green or Blue, and just write down your 12-word mnemonic.