Locks likely weren't that common anywhere -- I don't even think locks are as common now as they are in RPGs! (Much like big handy crates.) But padlocks are a relatively common find in archaeological investigations of medieval sites, and keys are depicted relatively frequently in medieval art -- here's a writeup on one medieval barrel lock, and another listing of ~12th century locks and keys found in an archaeological investigation of medieval York, and some more 12th and 13th century York locks.They would not have been ubiquitous across classes -- if you don't have much in the first place, why bother using a lock and key to protect it? -- and I doubt most instances especially earlier in the Late Middle Ages were sophisticated enough to warrant the obligatory lock-picking minigame instead of the intrepid adventurer merely smashing at the lock or breaking open the door or chest to which it was attached. (In the same way we now have incredibly sophisticated multi-level lock systems, but someone looking to steal a bike would rather cut a chain.) They were present and not unthinkably rare objects, or necessarily even especially mechanically complex. But they were functional enough as means of mediating access to spaces and storage that they remained in wide use.

Padlocks are extant from the Roman era onward, and clearly someone had to be making them -- a capable blacksmith could turn out (and can still turn out) a reasonably complex, reasonably protective lock. Locksmithing as a distinct trade is attested to by the 14th century, still closely related to blacksmithing but also the detail-oriented work of jewelers and goldsmiths.

For sources and images/reconstructions of high medieval and late medieval locks and keys, the Larsdatter page for padlocks includes a lot of extant archaeological examples, though the page for keys and keyrings/keychains has more instances from artwork than archaeology, possibly due to the materials and size of keys making them difficult to recover. (And maybe just because keys were a potent image of ownership and access in art/literature -- in the early 14th century, Dante uses the image of a lock and its two keys as a symbol of intimacy and access in Canto 13 of the Inferno.) There are still extant instances of keys as they would have been used, however, like this Tudor-era doorkeeper's key.

The developers mentioned in one of their videos that locks were not as common as shown in the game, and that they work differently than in this game. I would like to add to op's question: How did locks in 1403 Bohemia work?

Follow up question: how safe were medieval locks? It is pretty easy to pick most modern locks and we are much better today at fine mechanics.

At what point did it become common for the Chatlaine of a larger house have the larder, wardrobe, silver under lock and key?