Discrete log analog of ECM factoring algorithm? Anecdotally, most factoring algorithms have a corresponding variant algorithm that can be used to attack the discrete log problem using similar ideas.

Is there an analog of the elliptic curve (ECM) factoring algorithm, but for discrete logs?  Something that uses similar ideas, but can be used to compute discrete logs modulo $p$?
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.