"Emotional empathy" refers to the ability to understand and share the feelings of others. It means being able to put yourself in someone else's shoes and feel the emotions they are experiencing. Full definition
But there are downsides, including diminished emotional empathy. (adage.com)
The authors of this paper go on to highlight a few aspects of empathy, including what they call emotional empathy and cognitive empathy, an important distinction in academic work involving empathy. (positivepsychologyprogram.com)