> knowing that it's entirely coming from you and your experiences, and nothing to do with me and my own personality

That's not the point of empathy and not the point of my outreach. I don't need to know you precisely or be within a certain proximity in order to empathize with you.

> A year ago, I was talking to the local Catholic priest (I was donating some religious statues that I had effectively inherited), and it came up in conversation that I was going through a rough time. He went in for a hug, and it felt so absolutely empty and disingenuous.

For what it's worth, the man who did these things to me was a Catholic deacon, and the hypocrisy is blood-boiling. He would give very pleasant-sounding homilies about love, acceptance, patience and understanding, and then come home and savagely beat and torture me through physically painful punishments and extended periods of isolation. I would not go to a Catholic leader if you are looking for surefire genuinity. The institution attracts performative, power-seeking individuals.

> What do I say next? What's the next thing we can do in that interaction, or maybe a later one if I ever see them again?

It's a combination of bridging and bonding. Meeting individuals, like myself or a stranger on the street, and learning that you have something in common which provides substrate for conversation and communication through a shared experience, is bridging. Developing those relationships by building around that core is bonding.

We typically bond contextually: We both go to the same school or office and see each other daily, or we run into each other at the store each week, etc. I once ended up becoming best friends and living with someone who was my cashier at Trader Joes.

Instead of telling me our personalities and experiences have nothing to do with each other, we could discuss our experiences, find commonalities that are more than surface-level, and bond over those. I've met great people on this website. I've met some of them in person. Friends are all over the place, hiding in plain sight.