Hetzner

Respectable ones, do not store them, they just check them live. If they do, then they open the door for problems.

Because users need to be accountable for how they use their services, like hosting phishing websites, sending massive spam or scam campaigns. They would ask for a selfie, not stored, to match it with an ID.

Not all hosting providers do ask for ID. A credit card is not enough to prove identity because you can use virtual anonymous ones.

If the service has legitimate reasons, like it is a web hosting company or something governmental, then why not. If it is a social network, a game or the like, I will simply not use it.

Using services of a hosting company is something very serious, because the stakes are high, both in costs and in legal responsibility, so most likely you will need to prove your identity. Some companies will even require a meeting to sign a contract.