My impression is that in this regard, vetus is like the other third-declension adjectives "of one termination", such as fēlīx, atrōx, or -ns present participles. While adjectives like this are fairly uncommon, I don't think they're considered to be irregular just because of the way they (don't) inflect for gender.
I'm not sure what the historical origins of this conjugation pattern are, though, or whether vetus is actually like the other mentioned words from a diachronic perspective. Another oddity about the inflection of vetus, which it doesn't share with fēlīx and atrōx, is that it apparently is declined as a consonant-stem rather than as an i-stem in all forms1, but I don't know if this is at all related to the way it forms the nominative singular.
András Cser's The nature of phonological conditioning in Latin inflectional allomorphy (2015), which tries to give a synchronic account of Classical Latin inflectional morphology, contains the following footnote:
- Some adjectives do not mark gender in the singular at all (e.g., vetus ‘old’, audax ‘bold’). More precisely, they only mark gender by using the NomSing form for the neuter accusative (Masc/Fem veterem, audacem vs. Neutr vetus, audax). Note that in these neuter nominative-accusative forms the animate NomSing affix -s appears on adjectives like audax or all the -ns-final participles, which is very unusual from a systemic – and also from an Indo-European – point of view
The existence of the s~r alternation in vetus, veteris seems to be a sign that that from an etymological perspective, the "s" in the form vetus doesn't necessarily have to be identified as the suffix -s. The Wiktionary entry for the PI etymon "wetos" describes it as a neuter consonant-stem noun, like PI "genos" (the etymon of CL genus, generis).
I don't know enough to say whether Cser's analysis of the morphology of these adjectives (where the "animate NomSing affix -s" is interpreted as appearing in the nominative singular for all three genders) is correct from a synchronic point of view. Kyle Gorman's "Latin Rhotacism for Real" may be relevant, as it tries to give a synchronic account of various phenomena that are traditionally explained in terms of "rhotacism" of s.
De Vaan's entry for this adjective notes "P1.+; Enn. nom.sg. veter", so it looks like a masculine/feminine nominative singular form that was distinct from the neuter did arise at one point.
1. The unusual consonant-stem declension pattern of vetus, and some other adjectives that may show a similar pattern
Third-declension adjectives usually end in -ī in the ablative singular, -ia in the neuter plural, and -ium in the genitive plural, but with vetus we have vetere, vetera, veterum (at least typically; I don't know if any i-stem forms are attested as variants in the classical or pre-classical period). In this regard, its inflection seems to be similar to that of comparative adjectives (which have two terminations in the nominative singular: m/f -ior and n. -ius, and have the abl. singular -iōre, n. plural -iōra and gen. plural -iōrum).
There do seem to be a few more non-comparative third-declension adjectives that are not i-stems, a page from "Introductory Latin" on Louis Ha's website gives the following list of examples (all of one termination): compos, compotis; dives, divitis; particeps, participis; pauper, pauperis; princeps, principis; superstes, superstitis and vetus, veteris. Unlike vetus, many of these seem to lack attested nominative/accusative neuter plural forms in -Ca.
Charles E. Bennett's Latin Grammar gives a similar list, and mentions a complication with regard to dives: apparently no nominative or accusative neuter plural form based on the stem dīvit- is used (e.g. we don't find either *dīvita or *dīvitia; Lewis and Short also agree that forms like this do not occur) but we instead always find the contracted i-stem form dītia (for other forms of the word, it seems the contracted forms like nom. sing. dīs and gen. sing. dītis coexist with the uncontracted forms like gen. sing. dīvitis).
I have also found an old French source (Grammaire de la langue latine,
by J. M. Guardia and Justynian Wierzejski, 1875) that says that pauper, along with some other third-declension adjectives of one termination, is defective in the nominative and accusative neuter plural:
Beaucoup d'adjectifs à une seule finale n'ont point les cas de pluriel en a : puber, pauper, sons, suplex, trux, inops, memor, deses, etc., qui ne s'appliquent qu'à une seule personne. Dives fait ditia (et non divita ou divitia).
(My translation: "Many adjectives of one termination do not have the plural cases in -a: [...] etc., which only apply to a person (to one person?). Dives makes ditia (and not divita or divitia).")
I have just read the Allen & Greenough §122 page that you linked to in your answer to the question Are there any indeclinable adjectives?, which mentions inops as an example of an adjective where the non-occurrence of a neuter plural form is just a consequence of the meaning.