What is the general rule for classifying factor that only shift one supply/demand curve or two supply/demand curve?
Treating supply and demand as functions that relate the price and quantity for a good, a variable that appears as a parameter in both functions will shift both curves. In macro and micro theories, the details of the model will determine what these are.

When we consider supply and demand with econometrics, the fact that prices and quantities affect each other through two processes (are jointly determined)  results in endogeneity problems. Wooldridge's Introductory Econometrics has an excellent chapter on simultaneous equations models.