I think a short version of how this belief happened is as follows:
Democrats: "we believe this because we have empathy."
Republicans: "we disagree with your policy, despite your claims of empathy, because of X, Y, and Z."
Democrats: "you have no empathy because you disagree with our policy."
Some Republicans: "Have it your way. F*ck empathy."
Sounds about right.
Meanwhile, nobody takes a moment to pause and consider that having empathy perhaps just makes you a decent human being, regardless of your leanings.
People get so locked into their little boxes...