That would be a possibility: we can't say with certainty whether the wrong person is arrested. Obviously, there's embarrassment there, and there could be other damages. It's possible that the person may want to pursue civil charges. That's also why, in the context of citizen's arrest, we've added the “reasonable” period of time requirement. The farther away you get from the “finds committing” to the arresting at some point in the future, the greater the risk is that the wrong person will be identified.
It's part of the balancing act between what is reasonable and whether extending the time beyond “finds committing”, which is in the very act of the offence being committed, to some reasonable point past that, creates those risks. For the average shop owner, they may be less reluctant to arrest a day later than would a private security guard, who may have more confidence that they've identified the right person.
We're not foreclosing a person's civil remedies; if they are entitled to pursue those remedies, they will.