Are there any algorithms for a classification problem involving unlimited classes, and only a few instances per class The Scenario: A group of people must summarize specific parts of speeches they hear. They hear a new speech every day, and it's possible that multiple members of group are listening to the same speech.

The Question: What ml algorithm or approach could you take to link summaries that refer to the exact same part of the exact same speech?

Why this question is interesting: most classification problems deal with a large amount of instances and only a few classes. This problem is about dealing with very few instances (1-5) for a growing number of classes (The specific part of the specific speech).
Yes there are couple of methods. I recommend reading the Siamese Networks paper. From there you can check more modern approaches like Prototypical Networks and so on. In general the keyword you are looking for is few shot learning. 
All this approaches basically depend on finding similiarities between two or more inputs. One input is your unknown input and the others are your known samples. Your network will predict a class corresponding to maximum similarity.