At shift change we used to go around and let people know that a new bartender was coming on for the next shift and that they could either close out with us (they could reopen with the next bartender) or just leave it open. Most people will close out with the bartender that they started with and reopen later but not always.
Before I left the bar I would open up the POS and let the next bartender know about any open tabs. Regulars were always easy: “that’s Daphne’s tab, there’s Billy, do you know Curtis? he always wears a Cubs hat.” We used to add more details for non-regulars but we still kept it quick and descriptive. “That’s the guy outside in the Mets hat waiting on nachos” or “That’s the table of Aston Villa fans drinking pitchers. They’re friendly but they’ve been here since we opened”.
Table service was trickier (in sports bars at least) so we would track customers by what they were drinking and what they were wearing. Tracking table customers by seats in a non-corporate sports bar could be challenging unless everyone was watching the same event. NFL Sundays were more difficult than single sporting events because people in larger groups had a tendency to switch seats to check in on other games.
It’s been over a decade since I’ve worked in a bar that held onto physical cards but some still do this. At that location we had a lockbox behind the bar that was divided by letters of the alphabet. Mistakes were extremely rare but the two instances I can remember occurred when the customers had the same bank, shared a last name, and had nearly identical first names (think Robert and Roberta). One pair were siblings and the other were married. Overall it was pretty easy to avoid mistakes if you were paying attention.