All those private firms are in many ways identity providers just as real and official as governmental ones.
???
No they don't. They exist for the purpose of selling advertising. Any other purpose is either marketing copy to get you to use it or an emergent property based on people believing the marketing. Consider that LinkedIn would continue to exist if it provided no social capital whatsoever as long as it could still get ads in front of eyeballs.
Another observation: whether any specific social network "builds social capital" depends on the demographics of the audience and general "trendiness". People in high school don't care about LinkedIn, professionals in their 30s don't care about TikTok. Does this mean that TikTok should be an "identity provider" to people under 20?
Having said all that, I wouldn't want to use it as an identity provider. :)
The later one guarantees the identity: full name, date of birth, address, verified phone number, last taxable income, etc...
It allows to request government benefits or open a bank account online, because the identity is guaranteed. There is a real verified person behind the account. (corollary: you will be in troubles if somebody gets credit cards under your UK identity).
On the other hand, it's not great if that identity is required to apply to a job. The company can see your passport after they hire you. There is no need for every job board and recruiter and company to systematically get all your personal information in advance.