Should I use the same name for a member variable and a function parameter in C++?

That is correct, and allowed by the Standard. But a better approach is to use some naming-convention for member variables. For example, you could use m_ prefix for all member variables, then anyone could infer what m_state is. It increases the readability of the code, and avoids common mistakes.

Also, if m_state is the member, then you don't have to write this->m_state = state in the member function, you could just write m_state = state. In your current code, this-> part becomes necessary, without which state = state will become self-assignment.

Normally people just put an underscore after the variable or use shorter less descriptive var names for the function parameter.

I personally do not like the same name thing because when reading it, it is easy to make mistakes.