In the Soderbergh movie 'Contagion' the Matt Damon character is shown to have a natural immunity to MEV-1, the fictional virus that resembles COVID-19. Later on in the film he's shown flashing a wristband to a soldier which grants him access or the ability to pass. On the surface it sounds like a good idea.
Probably would be controversial. You'd have to give the govt. access to your personal health information. There's already trepidation about tracking people's movements by tracking where their cell phones have been in order for the authorities to have a better idea how the virus spreads. Also wonder if it would just be a simple plastic card or sticker for your DNI/CE, or a card with protection devices like holographic images and maybe a scanable magnetic strip so that it could be verified that the holder is who they say they are and the card would be linked to a database that confirms the holder is actually immune.
First thing that came to my mind was that there'd be a large demand for an identity card that "proves" the holder is immune to COVID-19 and thus gives the holder permission to be out and about. As Alan mentioned, some would be willing to intentioanlly contract the virus so they could get the card faster. Would put a greater strain on an already stretched thin health care system. Where there's demand, someone is going to create supply to fulfill that demand. I wouldn't be surprised to see black market counterfeit immunity cards. The way to make that harder is to make the cards more detailed and intricate instead of just a piece of laminated plastic or a sticker for your DNI/CE. But again, more detail and intricacies likely means the govt. having more personal information.
There's also no proof at this time that if you contract the virus and survive that you will then have immunity. From what I've heard the experts suggest that some past viruses have worked this way, but it's too early at this point to be certain that's how this virus, in it's current form, will be. Also, viruses mutate and change, so..... yet another thing up in the air about this whole situation.