My question: Is there any proof to say "Allaahumma Taqabbal" to your brother prayed at your side after performing the Salaat together or is this Bid'a?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

It is not a Sunnah to say ''Allaahumma Taqabbal'' to the person sitting next to you after concluding the prayer. This is neither reported from the Prophet nor from any of his companions . Even the four Imaams and others did not consider this as being desirable. So doing this as a habit and believing that it is from Islam, is an innovation. Among other innovations that the people are doing is shaking hands after concluding the prayer. Shaykh Al-Islam Ibn Taymiyyah said: 'Shaking hands after concluding the prayer is not a Sunnah, rather it is an innovation.'

Allaah knows best.