Yes, the speed bonuses do stack because they are from game features with different names
Bonuses generally do stack in 5e. Unlike in Pathfinder or 3e, there is no rule in 5e that bonuses of the same type do not stack. By default bonuses always stack, as long as they are caused by features that have different names.
This exception to stacking is on p. 252 DMG, Combining Game Effects:
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.
Because the Barbarian’s class feature is called Fast Movement and the Monk's is called Unarmored Movement, the two bonuses are not caused by game features with the same name, so they do stack, and your Barbarian/Monk will enjoy a speed bonus of +20 feet.