Is agreeing to a false claim in order to obtain financial compensation considered a false testimony, and what should be done about it?

Testifying contrary to the truth is a false testimony and a grave sin. The witness must repent to Allah for it. If this testimony resulted in your friend obtaining something he does not deserve, then you are liable for what resulted from it.

You must inform the concerned authority of the truth and seek their absolution regarding the financial compensation. If they forgive you, then that is well and good. Otherwise, you are religiously obligated to pay half of the compensation to them, as long as only you and the manager testified.