It was just a bad habit of Han's, to throw out nicknames or something like that with whoever he met. Poe wasn't that much younger than him, sure, but he'd also known a much older version of Han, which probably contributed to instinct. He didn't know about the others, but Rey and Finn seemed to be about Luke's age. So it worked just fine for him. And he was too stuck in old habits to really make much of a major change just yet anyway. From the sounds of things, it didn't look like he ever broke out of many of them.
"Shara Bey?" Han repeated. "No kidding. I thought your name sounded familiar. I fought with your father on Endor. Luke would know your mother better-" if he were there, "-they worked together a bit, but I have nothing but good things to say about your father. Makes sense, you being theirs, I can see a lot of them both in you." It was a whole-hearted compliment. They were crucial to the Battle of Endor and the few days that had followed. It was both weird and fascinating to meet their now-grown son. And, better yet, they were good people who seemed to love each other deeply - at least what little of the two Han had seen.
He didn't really feel that uncomfortable with Poe knowing anything about him, though. Plenty of people did. His reputation preceded him, or whatever. It wasn't a strange thing for him to encounter someone who knew him, or knew the Falcon, or knew what he'd done in this place or that one.
"I don't want to ask you anything too serious," Han shrugged, reaching over to get another drink of caf. "Doesn't seem like the time or place to have those conversations. Not that I'm not curious. But it doesn't feel real pressing right now either. So uh..." He looked Poe over, almost like he was analyzing him for a moment, then asked: "You ever fly the Falcon. You seem like someone I'd let fly her. You seem like someone who'd want to."