How Does Prime Factorization Break ECDSA?
However, to my understanding, the only purpose of Shor's Algorithm is quickly finding the prime factors of very large numbers.

Your understanding is incorrect. Shor's Algorithm is usable for both factoring integers and finding discrete logarithms.
Shor's algorithm works in two parts. First, it turns the problem (factoring or discrete log) into one of finding the period of a function. This first step is non-quantum. Then it finds the period using the Quantum Fourier Transform (QFT). Once you have the period of the function the first step can be reversed to find the solution to the original problem.
Shor's algorithm can be used without the Quantum part and simulating the QFT, though then it will be substantially slower than the best known classical algorithms. This python implementation simulates the QFT and may aid understanding.