Classify graphs using machine learning
The wording of your question makes it seems as if you only want to answer the question is the current graph to a specific graph A,B or C. If this is the case, machine learning might not be the best approach and in fact I would say your MSE based approach is probably a good starting place. If this is the case You may want to calculate the performance of this approach and see if it meets your needs. If you have a bunch of examples of graphs that are class A,B, and C then machine learning a probably good approach. 

If you have a bunch of examples of each class then you'll first need to calculate a set of features on each graph before you can apply machine learning techniques. After calculating the value of each feature on the graph you'll have a feature vector. From this point, the classification problem should be the same as the "single" data point problems you've already encountered. 

Example features:


Min or max x or y value 
The average slope of the series
Path length of the series 
Maximum curvature of the series
Coefficients from a polynomial fit


The tsfresh package will calculate a bunch of features for time series and help you sort out the relevant ones so its probably a good place to start. 

Good luck!