edit:
"Empathy is the capacity to understand or feel what another being (a human or non-human animal) is experiencing from within the other being's frame of reference, i.e., the capacity to place oneself in another's position."
https://en.wikipedia.org/wiki/Empathy
It's just the way that these words happen to be defined in English. They stand for different, but related concepts.
Why are you being so combative?
Other sources[1] define empathy as "feeling what another person feels" and sympathy as "having compassion for a person without necessarily feeling what they feel". In that case, the judge is clearly empathizing with the defendant, and not merely sympathizing.
People use words in a variety of ways - communication: a search for a language that is also yours!
edit: To be even a bit more contrarian here, sympathy in judges is something that I actively dislike, and why I'm a progressive in support of mandatory minimums, which is pretty much blasphemy. Sympathy in judges is why an affluenza defense can work, and why rapists sometimes get sentenced to probation.
Also the US has one off the most harsh sentences of any western country yet still one of those with most crime.
Even if a judge was softer because of sympathy he would still be much harder than most other european judges.
So not exactly anything that backs up your claims.
And empathy in one situation does not automatically mean empathy in all situations. Humans are funny like that.
It's not very hard to empathize with someone who you share similar life experiences with.
Real empathy would be, as you said, giving a shit about all the other felons that the judge cannot already identify with.
Here's my stance: I don't believe in cruel or unusual punishment. I believe that defendants who "remind me of my son" or "remind me of me, when I was a kid" or "me, if I had made one wrong choice" get better treatment than the heartless superpredators that don't look like the judges son.