To be honest, I used a third-party ID and selfie in 2023 and after a while they asked me to verify my face and since I didn't do it with my own ID, I couldn't solve the problem.
The only way to get out of that quagmire you got yourself into in the first place is to get that third party you used. If you still remember who you used and have their contact, go get them to do it and you will have the verification cleared it. Otherwise, there's no other way out of it.
... if I ask Binance that I want to be verified with my own name, will they accept it or will they block my account? any idea?
Your account will be cooked once you volunteer that information to Binance.