Its not just that. Its refering to people expressing extreme performative empathy, backed up self serving or useless actions. There are liberal grifters out there, just like there are conservative ones, and the phrase is used to label a person as one of them.
The way its used in conservative circles in the 21st and late 20nd centuries does mean that they see the empathy as performative and either self serving or naive, not that they think the person simply cares too much. Up until about a decade ago they might have even seen empathy as a noble trait, as long as it was backed up by personal action.