Some people say along with the mu'adhin.

Allah is the Greatest, Allah is the Greatest, and honor belongs to Allah, His Messenger, and the believers. I bear witness that there is no deity except Allah.

No, this is a mistake. You should respond after the mu'adhin as the Prophet ﷺ instructed, so say as he says.

Except when the mu'adhin says "Come to prayer," we say "There is no power and no strength except with Allah," and for the rest, we say as he says.

As for "honor belongs to Allah," this is not established from the Prophet ﷺ.