What is the term for learning to group objects
I would suggest you start with the Wikipedia articles on Categorization and more specifically Prototype theory.

While the Categorization article gives a number of branching points for you to drill down on and hone in and/or prime your thinking for further searches, the Prototype theory topic is where my mind went upon first reading your question

Even though the focus of prototype theory is on the shared artifact of language which you may or may not have been thinking of when you posed the question. I would expect any further reading you did on the topic to have plenty to say about the individual language user/learner also. 

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As a follow up I also found the following article which is from the perspective of Developmental Psychology.