As an experiment, I made a level 1 monk and level 1 barbarian character. I set all the stats to 18.
Monk's unarmored defense defines your AC = 10 + Dex + Wis. The Barbarian's feature adds Con.
Yet the AC was still only 18. Is this intended or is it a bug with dndbeyond?
As the other answer notes, different ways to calculate AC don't stack but rather you choose which one applies (generally the one that gives you the biggest number).
In this case however, there is a more specific rule: You actually cannot get the Barbarians Unarmored Defense by multiclassing, because the multiclassing rules specifically don't allow you to gain the Unarmored Defense feature twice. So you only have the Monk Unarmored Defense calculation available and couldn't choose the Barbarian version even if it would've given you a higher AC because you just don't have it.
Relevant Q&A: Can you switch to the other Unarmored Defense by multiclassing?
This is intended. D&D Beyond is following the rules from the Player's Handbook.
From page 164 of the Player's Handbook:
If you already have the Unarmored Defense feature, you can't gain it again from another class. (emphasis mine)
If you had taken the Barbarian class first, your Armor Class would equal 10 + DEX + CON instead, and you would keep your Unarmored Defense feature even when using a shield.
This is intentional. Both effects change the way your AC is calculated, and this wording is used in other effects. For example, Mage Armor causes your AC to be calculated as 13 + your Dex modifier, plate mail sets your AC to 18 (with no Dex bonus) and a breastplate makes your AC 14 + Dex modifier (max +2).
All of these effects are mutually exclusive for balance purposes, making a divide between effects that provide a bonus to AC (using a shield, Bracers of Defense, ect.), and effects that change how your AC is calculated. This would prevent you from using both even if the multiclassing rules didn't explicitly prevent you from gaining multiple instances of Unarmored Defense (hence making it so you cannot switch between the two versions of the same ability).
