An edit suggestion by "an anonymous user" just means that whoever suggested that edit wasn't logged in. Stack Exchange allows people to suggest edits even if they don't have an account; the link to do so will say "improve this question/answer" rather than "edit", but it works exactly the same way.
Judging by the content of that edit, it's indeed probably the OP, but for some reason they're no longer logged into their account. Normally, Stack Exchange uses HTTP cookies to try to keep users logged in, even if they used the "post as guest" feature instead of creating a "real" account, but this might fail for several reasons:
- The user may have configured their browser to clear all cookies at the end of each session, or they might be using some kind of a privacy extension that does that.
- The user might have been using Incognito / Private Browsing mode, which has the same effect of clearing all cookies at the end of a session.
- The user might be using a different browser entirely, e.g. if they asked the question at home but made the edit at work, or vice versa.
- The user might have originally logged in via Google / Facebook to ask their question, but forgot to do that again before editing.
In general (not speaking specifically about this site, but about Stack Exchange in general), I would personally say that it's usually OK to approve such edits if you're sure that it's indeed the OP, or if the edit makes the question objectively better. The only thing you really need to watch out for is accidentally approving vandalism. And even then, note that the OP (if they ever log back in) will get an automatic notification about the suggested edit, and can choose to revert it even if it has already been approved by reviewers.
If you're in any doubt, you could even consider approving the edit, but also leaving a comment saying that you've done so. That way, other users will be aware of what's going on, and can compare the two versions and potentially take both into account while answering. You can even link directly to the edit suggestion from the comment, e.g. like this:
I've approved a [suggested edit](URL here) to this question made by an anonymous user, who I suspect is probably the OP who just forgot to log in. If that's not actually the case, the actual OP may revert the edit if they don't approve of it.
Alternatively, if you think the edit is not a net improvement (and you can't turn it into one by improving it yourself), but you still suspect it might have been made by the OP and might contain some useful details clarifying their situation, you could also reject it but still leave a comment linking to it. That way, any useful extra information in the suggestion can still be preserved.
The one thing you don't want to do is reject a potentially useful edit without any comment, since such rejected edits will effectively disappear without a trace. (The review item will still exist, but it's only linked from the undocumented post timeline.) IMO, the only time you should be silently rejecting edit suggestions by anonymous users like that is if they're clearly unconstructive or vandalism.