It asks if you want to save the changes to the bin not to save the parts. If you say no then you are telling it not to save the changes which include things like deleting parts. You need to say yes so it saves the changes.
But you also say reasons for exporting files is to as backup and in that case you would not have the issue of it saying they are already exist because you are restoring things.
You also mention the clutter but for that you create new bins and sort your parts in bins. At that point you export entire bins to backup your parts and make restoring easier.
But my point was more that it is your actions that create the issue. If you didn’t save as a new part, export, delete and try and import again it wouldn’t be an issue and you may not even be aware the possibility of creating un-importable parts existed. If you simply used the editor as intended this bug would only come up maybe once a year at most. I can’t say I have had the issue except when you have edited one of my files manually and gave it back and it wasn’t importable. If you had imported the part, made changes in the editor, saved the changes and exported it Fritzing would have advanced the unique number (it does this every time you save the part in the editor) and that would allow it to be imported. But since you unzipped it, made changes and rezipped the file it was incompatible.
Sorry if that feels a little personal. It was not my intent to make anyone feel bad I just get frustrated seeing people making their lives harder when they could avoid the issues in the first place.