Anonymous edits are intentionally allowed by the system. The vast majority of visits to StackExchange are through anonymous sessions - meaning that the user either doesn't have an account or hasn't signed in.
The challenge, as you have hinted at, is that there is no easy way for regular users to link the anonymous edit with the OP's post. Because of that, anonymous edits need to be scrutinized carefully to make sure they're not adversely affecting the original question that was asked.
As a reviewer, there ought to be a threshold of improvement that an edit must provide before approving. Anonymous edits merely push that threshold up higher.
To answer your question - yes, anonymous edits should be taken seriously but there is a higher burden of proof with them. No, don't automatically decline them, but my experience has been that the vast majority of anonymous edits are rejected.
In this particular case, the OP is an unregistered account and likely used a different computer or forgot the password they used to create the account with. So there's a decent probability that the anonymous edit was from the OP or someone in the same lab. And yes, I did rely upon the mod tools to come to that conclusion. However, in reviewing the edit, I didn't see how it clarified the original question or made it better so I rejected it as "no improvement."