In practice and with respect to security, no, mathematically, yes. If you have a private key (N, D), there is algebraically an infinite number of solutions to the equation 1 = E*D (mod Phi(N)). However, if you make two such solutions (E, N) and (E’, N) that both satisfy the equation public, you will have compromised the secrecy of the private key.
However given one of the usual asymmetric schemes you can easily create such a scheme: To create a private key with n public keys, just generate n public-private keypairs in the normal scheme and define the “private key” to be the collection of the private keys. When signing just sign with all the private keys, when verifying try to verify one of the signatures. Encryption is the usual operation and decrypting should try to decrypt with all the keys (one of them should work).
Such algorithm is well known as “hierarchical deterministic wallets” and well explained here BIP-32