Is knowing the private key of RSA equivalent to the factorization of $N$?
Yes, they are (deterministically) equivalent. The original RSA paper (Section IX.C), working off Miller's results (Theorem 3), showed how knowing the secret exponent $d$ was probabilistically equivalent to factoring $n$.

Later, using more advanced techniques, Coron and May showed how to deterministically reduce finding $d$ to factoring $n$.