Create recommendation system to recommend products to a customer on any e-commerce website
To answer your 1st question,I am not sure if a recommendation system based on search history has been implemented or not.This approach does sound cool. 

Secondly, yes there are various algorithms to extract phrases from text. Phrase extraction and Text Summarization are one of the most important aspects in development of chatbots for conversational AI. 
To give you an example, there's a good library for this purpose, called PyTextRank.
Go through this link for better understanding of how this works.

Once you have a ranked summary of phrases for the given user's search history, you will be able to generate the list of item/s , that person is interested in. 
This is a good start for building the person's unique feature vector.
After this process, you can incorporate normal recommendation algos like Collaborative Filtering , etc. 

Hope this helps :)