They had to make up such explanations that stretch reality a little bit too much. Even in today's games, you can't log into a different game using the same account just because they use the same engine.
Moreover, even if it makes logical sense, it is far too convenient and unbelievable. It is like having coincidence after coincidence just to excuse Yui's presence in the plot.
Destroying suspension of disbelief does not require a completely illogical or impossible (story world wise) thing from happening. Just small parts here and there of hard to believe or way too convenient explanations will stack and are more than enough to cause it. The reason why I hate Yui so much is because she is a convenient tool to advance the story and make excuses for random events. She can do this or that, and it will "make sense" because her existence itself is already such a large leap of faith from the viewers.