Discrete log analog of ECM factoring algorithm?
Similar algorithms for factoring and discrete log are limited to pollard rho and sieve methods using a factor base. There are other factoring methods such as square forms, p+1 and ECM which as far as I know don't have logarithm-in-$F_p$-finding analogues. The factoring methods rely (explicitly for rho and implicitly for sieve methods) on treating a ring like a field, trying to measure the order of elements and waiting for the process to break down. Calculating a discrete log is a very similar process but it works correctly.
The other way to answer your question is yes there is a logarithm-finding analogue of ECM but it finds elliptic-curve logarithms. Similarly,the logarithm-finding analogue of Williams' p+1 method finds logarithms in a quadratic field. Not what you had in mind I guess.