HR policy are generally what you set up, subject to not violating the local laws. And of course, follow some rules if natural justice, fairness and best practices.
A practice of keeping a copy of an offer letter against which you have made a counter offer does not violate any laws, is not unfair or run counter to any best practices. So I don't see any anyone should have an objection.....
On the other hand, it makes eminent sense to have a copy of an important document for future reference and for any HR audit you may be facing.
You need to ensure the document is confidential, with access restrictions and not misused