If a monk hits a creature and stuns it with Stunning Strike—thus stunning it until the end of the monk's next turn—can the monk then use another Stunning Strike on the same creature the next turn, stun it again and stack the duration? My DM says the creature must first recover from being stunned before it can be subjected to Stunning Strike again.
Yes, you can stun a stunned creature
There is nothing in the description of Stunning Strike that excludes extending the duration. The feature says
When you hit another creature with a melee weapon attack, you can spend 1 ki point to attempt a stunning strike. The target must succeed on a Constitution saving throw or be stunned until the end of your next turn.
The Stunned condition also has nothing that would indicate that you cannot stun a stunned creature, that you can not attack a stunned creature (you even have advantage on attacks), or that it can not make Constitution saving throws. So all these things can happen, meaning you can hit and stun it again.
Stacking effects is explained on p. 252 DMG, Combining Game Effects1:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them - the most potent one - apply while the durations of the effects overlap. (...) Game features include spells, class features, feats, racial traits, monster abilities, and magic items. See the related rule in the "Combining Magical Effects" section of chapter 10 in the Player's Handbook.
So, the durations of the two stun effects will overlap, with the most potent one applying at any given time. In this case, they are all equally potent, so it does not matter which one applies while they overlap. Once the first stun effect runs out, only the second one remains, and continues to stun the creature.
1 This section is from 2018 Errata to the DMG, so it is possible that your DM's DMG does not yet have it, and they are unaware of it.