Do the double approval mechanism for payment and sanctioning of limits work in budget management and control?

I believe you are talking about the approval of payments process "double approval mechanism" is a sort of control for the sake of segregation and to review as a final authority by second level authoritative person.

In my opinion double approval mechanism has nothing much to do with the budget but yes It is a sort of control and can be a cross checking by two persons whether whatever they are signing are within the approved budget limits and if there is anything which is beyond budget and authorities doubt about can be questioned back to the originator of the payment and/or discarded.