1) They do not mean the same thing by ‘register’.
For Halliday, ‘register’ means a functional variety of language (with roots back to the Prague School and Firth).
Martin, on the other hand, equates ‘register’ with Halliday’s ‘context’, which removes the notion of register as a functional variety of language from the model. This is because a higher stratum is not a functional variety of a lower stratum; eg lexicogrammar is not a functional variety of phonology.
2) They do not mean the same thing by ‘context’.
For Halliday, ‘context’ is what people do with language (H&M 1999: ix), the ‘semiotic environment’ of language (and other socio-semiotic systems such as image systems (p375), the “culture”, considered as a semiotic potential (p606). Halliday’s ‘context’ is a level of abstraction that is realised by language.
For Martin (1992: 496), ‘context’ (register and genre) are levels associated with ever larger ‘units’: just as the level of discourse-semantics tends to focus on an exchange or “paragraph”, the level of register tends to focus on a stage in a transaction, and the level of genre tends to focus on whole texts. To this extent, Martin’s ‘context’ refers to levels of abstraction of texts.
