There are a number of scripts which can help with this process - some require batch save processes, some do not; you could, I'm certain, with more time than I've put into this (<5 minutes including writing and editing this response) find more specifically relevant data on the Adobe fora.
Removing metadata from all images
Excerpt from linked forum article above:
The best third party software is ExifTool
http://www.sno.phy.queensu.ca/~phil/exiftool/
If you wanted to do this in Photoshop you could use Picture Processor to batch save the pictures as Save for Web selecting No Metadata
http://www.scriptsrus.talktalk.net/PP.htm
There are other methods that could be used, you can create new JPGs from the cache in Bridge.
http://www.scriptsrus.talktalk.net/BridgeProcessor.htm N.B this one is Windows Only!
How to replace metadata in Photoshop
Excerpt from linked forum article above:
A script can be found here to strip Exif metadata by creating a new document with the same pixel data and filename:
strip exif
You could add code into this script or even record an action (screenshot below) for the addition of the copyright metadata after running the script. This could be run as a batch action or used in Image Processor for batches.
action.png
Or you could try posting in the Bridge Scripting forum to see if there is a way to do this via Bridge…
However I would just recommend using ExifTool. If you are on a Mac, one can incorporate the ExifTool CLI code into an Apple Automator Service so that all you have to do is contextual right click on a folder or file.
Hope this helps with your issue.
BTW: @Luciano is correct that his answer is indeed helpful - he is telling you not just about Photoshop's default behaviour, but in fact its designed behaviour, which is a helpful and valid answer to your query, especially given that we as a community are graphic designers, not coders or even privacy activist coders. What you are looking for is either a script or code approach or an actual hack of the software - in either case pretty darn close to out-of-scope as a question here on GD.SE.
If you end up not feeling satisfied with the responses you get here I'd suggest you reaching out to Adobe directly, as they (one would hope) must have encountered this issue before.
Note: you've now been provided with several workflow-based approaches to address the specific need, which is what we here at GD.SE are pretty much all about. Hopefully you get more such answers, and one which so closely matches your needs that you can move forward. If however, you personally will not allow for realistic workflow changes, but rather are sticking at requiring a re-coding of a high-capital-cost software product which is the flagship product of a major software vendor, that will be definitively out-of-scope here.
If someone here knows of a plug-in which will anonymise EXIF, perhaps we'll ALL get lucky and learn about this, which is I think the best-case scenario.
Good luck - I sincerely hope we find you an answer to this issue.