AFAIK there are two separate quantum Shor algorithms, one for integer factorisation and one for discrete logs.

https://cds.cern.ch/record/602816/files/0301141.pdf

I'm not familiar with the intricacies of the algorithms, but there is also the consideration that problems that are of a similar difficulty from a classical perspective are not necessarily equal for a QC, due to the entanglement of qubits meaning that processing power scales 2^n.

But I'm not sure that for a QC the distinction between ECDSA difficulty and RSA difficulty is really much of an issue. If something is built that can break ECDSA, then RSA will not be far behind. All asymmetric cryptography is theoretically vulnerable.