When can a museum be considered an academic institution?
My answer is "localized".


In France there is a distinction between museum of arts (arts in a broad sense, antiquities being arts for instance) and museum of natural sciences (biology, paleontology, etc.). The latter is considered an academic institution: part of the people working there have the title of associate or full professor, and they can have PhD students.
In some european countries (France, UK, belgium at least), some museums are part of a university, so I think we can say that they are academic places.


The real question may be then "are you an academic if you work in an academic institution", and I guess the answer is no.