Is Karp Reduction identical to Levin Reduction
No. First note that Levin reduction only makes sense with respect to classes which certificate has a meaning, e.g. $\mathsf{NP}$ while Karp reduction is general. The word "certificate" for a problem makes sense only when a verifier is fixed. Levin's reduction assumes that the verifiers are fixed. You cannot change the verifiers. (In the following I assume that certificate verifiers are fixed as required by definition of Levin reduction.)

Second, Levin reduction means that one can find the certificate for one from the certificate for the other. It is true that the well-known Karp reductions between natural problems turn out to be Levin reduction but this does not need to be true in general.

If we can reduce instances of a problem $A$ to problem $B$ it doesn't mean we have a way of computing a certificate for one from a certificate for the other.

For this to be true we need the fact that a certificate search problem corresponding to the decision problem is polynomial time reducible to the decision problem. This is true for $\mathsf{NP\text{-}complete}$ problems but not known to be true generally even for $\mathsf{NP}$ problems.