Since, in the time of the Prophet (ﷺ) and his companions the madh-hab did not exist, could we classify madh-hab as bid’ah in Islam?
Could we classify madh-hab as bid’ah in Islam? No, because there are many other things which did not exist at the time of the Prophet (ﷺ). We are not talking about innovation that is outside of the deen. We are talking about within the deen, in this regard.
The Prophet (ﷺ) used to say, when beginning a Khutbah (sermon): "… The best of speech is the Book of Allah and the best of guidance is the guidance of Muhammad. The worst of things are those which are newly-invented, and every innovation is a going astray…" 
Ayesha Bint Abi Bakr narrated that the Prophet (ﷺ) said, “Whoever innovates something in this matter of ours that is not a part of it will have it rejected.'" 
Concerning the madh-hab, it only represents a school of thought. After the death of the Prophet (ﷺ), the Sahaba taught Fiqh. You would find in Makkah, certain Sahaba teaching and some were teaching in Madinah, while some were teaching in Basra. They taught in different places. Those who stayed around them, learned under them, you could say that they were the students of the madh-hab of that Sahabi.
However, they did not use the terms, but it is known. Those who studied under Ibn Abbas conveyed or transmitted his Tafseer of Qur’an, etc. and his opinions, etc. If you wanted you could call it the Abbasi madh-hab. You could, but they did not use those terms. The terms came later.
So it is not really the name, rather it is the concept. The concept of scholarship had being passed down from generation to generation. This concept is a part of Islam, andit is not outside of Islam. Giving it a name does not make it bid’ah. What makes it bid’ah is where people become fanatical about it. So much so that they then say, we cannot even pray behind each other. This now is bid’ah. Where they say that you cannot marry a Shafi’i- this is bid’ah! Therefore, this is where it ends up in bid’ah.
 Sahih Muslim: 867
 Sahih al-Bukhari: 2697< Back to Questions