It depends on what you mean by “service”: I am not interested in trading cryptocurrencies, so I will never need that kind of service. I just need to accumulate BATs, and it works.
Like I said, Gemini verified my account right away (and it was night in the US).
I guess people who don’t have a passport, or who try to send useless (not to say fake) photos, get blocked.
I can’t find any sources claiming that the only purpose of BATs is anonymity. Not even bitcoins are anonymous. If you need anonymity, you should use other cryptocurrencies, such as Monero.