We have various private companies taking copies of our ID; in the UK, you'll have scans of your passport/driving licence taken for various reasons.

We shouldn't have to trust them to get it right; and I suppose the threat for them is fines if they don't, but it's not good enough.

Probably, a more solid solution would be to offer a government ID service where these companies check against a central database that already holds your information and then they have to keep nothing.

If I hire a car, I provide the rental agency a code that gives them temporary access to my driving record to ensure it meets their requirements, it's a one-time code and I request it when required and provide it to them; something similar could be adopted for other purposes if they have a legal requirement to verify your identity.