Please forgive me if I'm jumping to conclusions here, but I'm guessing you're American.
> What happens if I don't want to identify myself?
In some countries that is not an option. For example, in Portugal when in a public space, the police has the ability to ask for ID if they have a reason. They must first identify themselves, and list the reasons to ask for identification. If you have no valid ID with name and photo, someone that knows you and can attest to your ID can do it, IF they themselves have ID. If you're alone, you can either ask someone to take your documents to where you are, or ask the police to go with you to where you have them. If that is still not possible for whatever reason, they you can be taken into a police station until you're properly identified and your fingerprints are taken.
After all that, if the reason why you were identified lead to nothing you can request for the info on you to be eliminated.
> What happens if poverty does not allow me to carry a digital ID?
Most countries have state issued ID cards. Note, I wrote most, not all.
> What happens if I lose my digital ID wallet or it is compromised?
Same thing that happens when you lose your wallet, I guess. Go to the nearest police station, tell what happened. Then go on from there.
> Also judging by dealing with the average citizen on a daily basis, even securing an iOS device and remembering the iCloud password is beyond their level of care.
People take government stuff more seriously than they do other things. For example, my mother that can't remember her router password knows her hexadecimal password to the IRS website by heart. My in-laws kept the letters from the IRS with the password in a well known place just to use it once a year when they need to fill their forms.
I'm not saying there's implicit trust in governments in the EU, but from the discourse on the internet, it's clear to me that most (western) european citizens trust their government more than Americans, overall.