Considering fake vaccine cards are already making the rounds, I doubt the positive test is easier to fake.
Looking at my vaccine card, there is nothing special like a ID number or even a signature from a doctor.
The stickers for each shot have the name of the vaccine and a lot #, so that might be the tough part, but the lot numbers are screwy. The first one is a ###A##-#A sequence and the second one is a ###A##A sequence. Not sure if there is a national tracking database on that part. No barcode for easy scanning so it probably isn't checkable.
My Dad's testing cards (The VA was doing a test for every appointment so he got tested three times in two weeks because he was having a non-COVID related issue) had a doctor's signature and the results.
Wouldn't this apply to getting a vaccine, what makes tracking a diagnostic test so much more difficult? People could just have a photo of their positive test, just like they have a photo of their vaccine card...