This requirement isn't about whether or not New Hampshire considers other state driver's licenses valid for the purposes of assessing someone's ability to drive, but enforcing the requirement that a person cannot claim primary residency in more than one place. As far as I know, it is a requirement in every state that you declare that state your primary residence in order to receive a driver's license. As a practical matter, this makes it more difficult to register to vote in more than one place, and makes it easier to determine the best place to call you for jury duty, among other things. As part of that declaration, the state issuing the license is requiring that you surrender your other license because it prevents you from trying to use it to defraud the other state by claiming you are still a resident there. As was pointed out in comments, the states don't necessarily communicate to each other that a license was surrendered, but in order to get another copy you would have to certify that you are primarily a resident of the other state again, which would be perjury.
This doesn't really have to do with the Full Faith and Credit clause, since the states aren't refusing to accept the records of the other state. To the contrary, the few times I've changed my license I can use my old state's license to prove my ability to drive - the new state puts their faith in my proper testing by the old state, rather than forcing me to retake a test (in fairness, they also accept out of country licenses of their own volition, having to redo tests for everyone moving to the state would be a huge drain on resources for little gain). For what it's worth, while it seems like states usually ask that drivers change to the state's license within x days of moving to the state, I've never actually seen a penalty attached to failure to do that, and for the purpose of certifying your ability to drive valid out-of-state licenses are always accepted, even if there were a penalty for failing to change them after you move to the state.