Let's have a look...

> and a private key akin to the signature

Nope, a private key is not akin to a signature. It's akin to a... key? Or perhaps a special pen / wax-seal that you use for signing? I don't think there is an exact analogy?

> To carry out transactions, those codes are passed between bitcoin wallets

Lol, nope. Private keys are not passed around. Neither the public keys (assuming they mean 'addresses' here), which are actually hashes of the public key.