There is really nothing specific to API layer (I am assuming you mean REST API layer) in building backwards- and forwards-compatibility compared to databases. If anything, it is less powerful.

It is pretty easy to do with databases as well, you just need to adopt the right mindset.

For instance, if you think having an "api/vX" of an endpoint is acceptable, then it must also be to create a duplicate table/relation — you'll have exactly the same challenges in maintaining consistency between the two, though RDBMS offer quite a bit of tooling built-in.