Sure, you can prefer to use subsets of a single time series instead of multiple time series. The issue remains that it doesn't matter what your performance on training data is. You still need to partition your dataset into training and test data - otherwise you could just be storing a lookup table for all you know. It looks like the author has trained on the entirety of the dataset, and then is just considering that performance...
Let me put this another way. You do this with a random walk. Train on your entire timeseries - every length 50 window. Say that there's only 8 unique values at each timestep. That means that there's 8^50 possible input sequences into the neural network. A sufficiently complex neural network can fit an arbitrary function, so if you just have a couple thousand windows there's ~(8^50 / 1000) possible functions that can predict the correct output exactly - and this is on noise! And in all likelihood the neural net will learn that noise: https://arxiv.org/abs/1611.03530 Without comparing training and test results there's no way to know that neural network learned anything of value - it can get perfect accuracy on training data that's pure noise!
This stuff is really critical to get right if you're doing machine learning.