Earlier quoted context omitted.
I was going to say the same, but it looks like the documentation mentioned is for the client indeed. If the client is MIT licensed, are you infringing on anything by writing a compatible API?
That question is the essence of the ongoing Oracle v. Google case: https://en.wikipedia.org/wiki/Oracle_America,_Inc._v._Google... .
Moreover, the court ruling makes it seem that if the intent is interoperability, fair use may apply (only the court rules Google's intent was not interoperability, as their implementation was intentionally incompatible).
Although in this particular case, it appears they can claim copyright violation on the documentation.
[1]: Most notably, in order to be copyrightable, a work has to be "fixed in a tangible medium of expression" (https://www.law.cornell.edu/uscode/text/17/102), i.e. you need to have a specific text (or image) that you can say, this is the work (although then even derivatives are protected). This is true for APIs, but not for protocols (or REST "APIs"). Whether this distinction makes sense to programmers or not is irrelevant. The same distinction holds for programs vs. algorithms: programs are "fixed in a tangible medium", and are subject to copyright, but algorithms are not, and not subject to copyright (but can be protected by patents).