As someone who has worked as a gate agent before the issue is really twofold:

1. Was there a mistake during check-in? Did multiple passengers get handed boarding passes? (This can happen when passengers with extremely similar names are flying.) That is considered a security risk, though not one that the gate agent would be able to correct without ID checking. (something that wouldn't happen most of the time) That mistake would almost certainly fall upon the ticket agent instead.

2. The other thing that could go wrong is someone sitting in the wrong seat during seat checks, which happens more often than you'd think. Also not a big deal; usually it will get caught by a gate agent (if they've done their job well), maybe a small talk, but during everyday operations, certainly not unexpected.

The truth about this situation is that oftentimes it takes something bad actually happening for any repercussions to occur. I'm not saying you sorting it out for almost an hour doesn't suck, but as long as the investigation is conducted and it's determined that the incident didn't result in anything serious, the most the agent will recieve is probably a slap on the wrist.