Is each form of word classification also considered to be '(named) entity recognition'?
Is this a fair assumption?

No: Named Entity Recognition (NER) is a specific task which consists in detecting named entities. The more general term for this kind of task in Machine Learning is sequence labelling, because it's not only about classifying words but annotating a sequence of instances in which order matters (e.g. words).
It's true that NER is certainly the most famous task of this kind, but there are other important ones, for instance Part-Of-Speech (POS) tagging.