The impact on game balance would be minor
Both features set your AC, and any gains from switching would be minor — mostly the benefit from switching to barbarian is that you can use a shield. And while +2 to AC is a real boost, it costs you the use of one hand. It would be an upgrade, because the barbarians AC, if optimized, can be one of the best in the game.
There is also an indirect penalty from this multiclassing approach vs just going Monk or Barbarian straight on, in that you would want 3 high Attributes - Wisdom and Dex for the monk, Constitution and Dex for the Barbarian, and unless you rolled your attributes and got lucky, it is hard maximizing even two, never mind three. So you'll have to pay with worse AC earlier on if you skimp on Wisdom, or later on, if you skimp on Con.
I think from a balance standpoint, it would have been possible to instead say "If you already have the Unarmored Defense feature, you can choose which one to keep." I suspect the reason to avoid that is to keep things simple and reduce bookkeeping, which was one of the design goals of 5th.