Use empathy if you're looking for a noun meaning "the ability to identify with another's feelings."
When Bill Clinton famously told people "I feel your pain" during his 1992 election campaign, some praised and others ridiculed him for displaying empathy, the sharing or understanding of feelings. Empathy is different from sympathy, which is pity or sorrow for others' misfortunes. They share a common root in -pathy, from the Greek pathos, "feeling." Where they differ is in their prefixes: sym- means "with," while em- means "in." If you can empathize with someone, it's because you have been in their place: you've "walked a mile in their shoes," as the saying goes.