Two is a paltry number, barely plural, & a two-point scale left to its own devices needs only to distinguish before it can put its feet up: it's otiose to muse on whether equal intervals or equal ratios are meaningful when there's only a single interval or ratio to consider, or on whether ranking is meaningful when there's only one sequence a pair can have; all the operations you might want to perform on the data are unaffected by its representation, as @Tim has explained.
It's only for the external relations of a binary variable that these things matter at all. The Jaccard index is a measure of similarity between two individuals each having several attributes represented by binary variables; you calculate the ratio of the number of attributes for which both have "1" to the number of attributes for which either have "1". Clearly the coding as "0" & "1" isn't arbitrary here (though we could swap it round for all variables at once & make a corresponding change to the calculation of the Jaccard index). This is the situation in which @ttnphns talks of "ordinal dichotomous variables", which seems fair enough. An example can be found in Faith et al. (2013), "The long-term stability of the human gut microbiota", Science, 341, 6141, where the Jaccard index is used to measure the similarity of the make-up of an individual's gut flora at different time points—the ratio of the number of bacterial strains in common over the total number of strains found. The choice of metric seems sensible—why take into account all the different strains absent at both time points? could an exhaustive list even be compiled?
A more hum-drum example might be found in the various ways variables are often combined into indices, scores, or whatever; to serve as, say, descriptive statistics, or predictors in regression. To calculate the Charlson comorbidity index you add up dichotomous variables that indicate conditions such as myocardial infarct & congestive heart failure. Many conditions are coded with "0" & "1"; but as hemilplegia contributes 2, & malignant tumor 6, to the total score, I'm tempted to propose these as interval-scale dichotomous variables.
Needless to say, how you align different binary scales in these kinds of situations depends on making decisions appropriate for the job at hand rather than somehow intuiting the true nature of each individual scale—an attribute coded "1" for the calculation of one Jaccard index might be coded "0" for the calculation of another.
The paragraph above exemplifies something that's always the case with this business of scale types. Stevens points out various relationships between which features of how you represent data need to be considered meaningful & the kinds of operations you perform during your analysis:
Scales are possible in the first place only because there is a certain
isomorphism between what we can do with the aspects of objects and the
properties of the numeral series. In dealing with the aspects of
objects we invoke empirical operations for determining equality
(classifying), for rank-ordering, and for determining when differences
and when ratios between the aspects of objects are equal. The
conventional series of numerals yields to analogous operations: we can
identify the members of a numeral series and classify them. We know
their order as given by convention. We can determine equal
differences, as $8-6=4-2$, and equal ratios, as
$\frac{8}{4}=\frac{6}{3}$. The isomorphism between these properties of
the numeral series and certain empirical operations which we perform
with objects permits the use of the series as a model to represent
aspects of the empirical world.
This is an instance of an important general principle: you don't want arbitrary or conventional decisions about how to write things down to materially affect your conclusions.
The type of scale achieved depends upon the character of the basic empirical operations performed. These operations are limited ordinarily by the nature of the thing being scaled and by our choice of procedures, but, once selected, the operations determine that there will eventuate one or another of the scales listed in Table 1.1 [nominal, ordinal, interval, & ratio].
So you can't, for example, average scores on a five-point scale and claim that the interval between scale points doesn't matter: something's got to give (& note that it may well be the claim rather than the averaging—see e.g. here). It's a mistake to confuse this prohibition with the stipulation that first you need to determine the true scale type & then think about suitable methods of analysis. See Should types of data (nominal/ordinal/interval/ratio) really be considered types of variables?.
ordinal present absent
. $\endgroup$seems to give some weight to its values
is relevant. $\endgroup$