X509 converted to Encrypted and Signing keys We have proprietary tool which takes a p12 file and generates a .key file. Inside the key file, I see two PEM formatted sections: Encrypting Certificate and Signing Certificate. When I open them on my machine they have the same CN, Expiration and Issuer, etc. 
This key file is used for mutual 2-way authentication and we provide the key file to the client. 

What is happening here? I understand the regular mutual authentication where you and client exchange public keys and trust them but why do you need to implement this method?
The PKCS12 (p12, pfx) is a standardized keystore where you may store multiple keypairs (key and certificate) or certificates themselves. It can be read/manipulted by multiple tools (OpenSSL, Keytool, keytool-explorer, ...).

You may have multiple certificates in you PKCS12 archive, it's usually a certificate chain with the certificate itself and its issuer (certificate authority). The certificates should be 'chained' (in a hierarchy).

Mutual SSL is used where the trust needs to be mutual (it's not only you want to be sure you speak to the right server (server only SSL), the server wants to be sure the client is who it claims to be). During the SSL handshake (or SSL renogotiation) the SSL server passes a set of trusted certificate authorities and the client then provides its own certificate if it is trusted by one of the offered CAs. Then the identity (private key ownership) is ensured by exchanging encrypted piece of data. It is a very effective method ensuring that only trusted clients are able to connect to the service.

Updated: why are there two certificates with the same user data but different purpose?
You have 2 certificates to the same key? Do they have the same issuer? Technically any certificate could be used to authenticate or to sign data. Legally - it may be different. It is common (e.g. on the eID cards) to have separate certificate for authentication and for signing. So you don't sign anything (legally binding) while you think you are authenticating to a service.. But -if they are both in the same keystore - don't you have the certificate chain there?