Training a model sample by sample
In the general case this is impossible because standard learning algorithms require all the instances at once in order to compute the model. They usually work by making calculations based on the set of instances, then only the results of these calculations are stored as a model (for instance a linear regression classifier stores only the final weights which are needed to make a prediction). 

It is however possible to train a model incrementally:


Very often this is done by simply adding the new instances to the original training set, then re-train a new model on the full set of instances (new and old ones together).
There are also actual incremental learning algorithms, including adaptations of some common learning methods. However this is not very common, I'm not aware of these being implemented in the most standard ML libraries that I know of.