Storing the dislike counts for each user takes a binary state at the minimum, and a byte of storage typically. Considering the HUGE global user base, that would be too much data storage and maintenance required for a company like Google/Youtube to invest their resources in. It also improves user experience by minimizing the data transfer requied to consume products in the platform, thus improving the efficiency and speed in the platform.

It is for the best of the users, and is a great design for the user. User's will learn to love them in time.