This idea was first studied by Shanks, Pollard, Atkin and Rickert, although they didn't write a paper as far as I know. Schnorr and Lenstra gave a heuristic time complexity analysis in their 1987 paper. Since there's no standard name, I'll call it the class group method CGM.

The time complexity of CGM is soft-O of $L_n[1/2]$ meaning we drop logarithmic factors. There's a second constant in the time complexity but its not a strong predictor of an algorithm's practical performance so I'll ignore that. This time complexity is like all algorithms for factoring $n$ from that period with the sole exception of ECM. The time complexity of ECM is soft-O of $L_p[1/2]$ where $p$ is the smallest prime factor. Note that $n$ doesn't show up in the time complexity. In practice, the size of $p$ is the most important factor in the algorithm's performance.

There have been [attempts][1] to make CGM more competitive with ECM. The challenge there is that for ECM the practical performance is controlled by the size of the smallest prime factor of $n$ rather than the number itself (just what the time complexity would led. This is not true for CGM. 

However, in the special case that $n$ is not squarefree there is [new work][2] on an algorithm that improves on CGM. Perhaps with some further work this algorithm's time complexity will depend on the size of the squarefree part of $n$. On the practical side, the author gives some evidence that the algorithm is fast compared to the best factoring libraries available for numbers of the form $n=pq^2$ where the primes $p$ and $q$ are of a certain size. 

This algorithm takes advantage of properties of class groups of non-maximal orders. Extending it to a broader class of $n$ would require new ideas. 


  [1]: https://prism.ucalgary.ca/items/21d05d08-9aab-4b28-8933-ace085804821
[2]: https://antsmath.org/ANTSXVI/papers/Mulder.pdf