The place I sent my parrot to when he passed puts a numbered metal tag on each pet before they go in and the tag stays with the ashes. They also did a foot impression and sent the ashes back in a simple but lovely clay urn. It was a little weird seeing the ashes in a ziplock inside the urn but I totally get it since they were a small family business.
This is like the character Orlando in Tad Williams’ Otherland series.