Why can’t the same RSA key be used for identification and encryption?

Alice and Bob use an identification protocol, in which Bob chooses a random number r, encrypts it with Alice’s public RSA key and sends it to Alice. Alice replies by sending r back to Bob. Why can’t the RSA key be used for message signatures or encryption?

If there was only 1 key, it would have to be the public key, therefore everyone could access it, encrypted or not. With a public and private key, it is secured. You can encrypt a file with Alice’s public key, but only Alice with her own private key and decrypt it. The RSA algorithm involves three steps: key generation, encryption and decryption.