It's absolutely enmeshment. I think enmeshment can emerge when children with higher than average levels of empathy aren't given the tools and support they need to deal with that.
(Also now a psychologist, so I could be off base. But they do at least seem to be related things.)
Sounds like you are confusing https://en.wikipedia.org/wiki/Enmeshment with empathy.
Disclaimer: I'm not a psychologist.