The house and its seat
Blochmann’s register of Akbar’s grandees enters the head of this house as “Ra’jah Biha’ri’ Mall, son of Prit’hiráj Kachhwáhah”, and records that there “were two kinds of Kachhwáhas, Rájáwat and Shaikháwat, to the former of which Bihárí Mall belonged.” Their seat was Amber, in the Mughal province of Ajmer.
The same page carries an observation that is easy to skim past and is the whole reason Amber mattered: “Though not so extensive as Marwár, the revenues of Amber were larger.” The house that entered Mughal service first was not the largest in Rajasthan. It was the richest per acre.
What the alliance actually was
It was not one event. Blochmann’s account has a sequence: a request to enter service, a matrimonial alliance granted, a daughter received at Sambhar, the Rajah and his son and grandson joining the emperor afterwards, and only then a rank — Bihari Mal was made a Commander of Five Thousand at Agra. The marriage sits inside a service arrangement rather than standing on its own.
The direction of that arrangement is its most durable feature, and it ran one way. Amber gave a daughter and received rank, office and protection. Nothing in either source describes a movement in the opposite direction.