The action API allows setting the user language with the uselang parameter. This is distinct from the API's own "interface language" (ie. the language in which, for example, error messages are in), and influences content returned by the API (primarily, it influences how the MediaWiki parser behaves, and how MediaWiki interface messages are looked up). The difference is most relevant on multilingual wikis like Commons where the content itself significantly depends on which language you are reading it in, but API calls which are used to create user interface dynamically also rely on this.
In practice, uselang is used to override MediaWiki's internal user interface language setting, which is used by the i18n system, the parser and a couple other things. The action API framework does this (in ApiMain) by setting the language of the main RequestContext, which is then queried by various MediaWiki classes in various places in a hard-to-predict manner (which is unfortunate but something we'll have to live with for a while).
How should this happen in the REST API? One option is to leave it to the handler to define the language as an explicit API parameter, and then set it. But 1) this would mean that every handler would have to predict that it has a dependency on the language, and declare it (arguably not a bad thing), 2) this would mean that anything that depends on a service that takes a language parameter (and gets that from RequestContext::getMain() at service instantiation time) behaves unpredictably as the language will depend on whether the service got instantiated before the handler was called.
(In theory, ServiceWiring should not use RequestContext. But also, in theory, services should not directly call RequestContext::getMain() themselves, which is static coupling and makes unit testing hard. There is no way today to avoid both and I don't think there's much consistency in which one a given piece of code prioritizes avoiding.)
So, there would be value in providing a signal (a HTTP header, maybe; or it could rely on the user's language for authenticated requests) for setting the interface language early enough that it's guaranteed to be consistent during the service setup phase.