I don't think so, it is either auto number or text in which case it'll be required, can not be left blank, salesforce autopopulates the record with the id in case you do not provide one and there is no setting to change that behavior (that i know of on current releases)
I'll recommend changing your requirements and at least populate the name of the record with something, a naming convention for the record on it's current stage would be helpful,
That way you can list all of your "ANONYMOUS" records and assign them into a queue so a business person can go and put the right info just to cite an example, a custom unique identifier which can be recognized by your system or an external application, but never blank