This is a great point, that register is not a type, so as Bill says there is no such thing as 'a register'. It's far more useful to think of field, tenor and mode as register variables.
On the other hand, a genre is a type (which is what 'genre' means). A genre is defined in Sydney School research as a configuration of register variables, that 'persist as recognisable types within the culture'.
 On the SFL model, the term 'register' refers to a point of variation on the cline of instantiation, as viewed from the system pole; viewed from the instance pole, that same point is termed 'text type'. Registerial variation is modelled as different probabilities of feature selections on linguistic strata. Such probabilities are established and altered, in the long term, by the actual frequencies of such feature selections in the instantiation of (actual) texts.
 It is not 'far more useful to think of field, tenor and mode as register variables' because doing so confuses context (field, tenor, mode), which is more abstract than language, with register, which is functional variation of language itself (not of the semiotic context that language realises). That is, it confuses stratification with instantiation. Registerial variation is a feature of all linguistic strata.
 It is precisely because 'genre is a type' (inter alia) that it is anomalous to model it as a stratum — as Martin does — in SFL. The other strata are not types: semantics is not 'a type', lexicogrammar is not 'a type', phonology is not 'a type'. Functional variation of strata is modelled by (points on) the cline of instantiation.
 On the 'Sydney School' model, a genre is not defined as a configuration of register variables. As Martin's stratification construes it, genre is realised by register; that is, the model construes genre and register as different levels of symbolic abstraction.