Mubi is the best described member of a small "Mubic" subgroup consisting of Masmedje, Kajakse, and Zerenkel. The other members are much more poorly described, but enough can be gleaned from published materials to show that they share essentially the same system of subject pronominal suffix series with different syntactic functions.
For Masmedje, a short text is provided in Marti et al. (2007), from which much can be gleaned using Jungraithmayr's Mubi grammar. Sentence 3 is particularly illuminating:
wa ka-sek-a an-sual-ukun and[?] 2-come.PFV-? 1Pl-ask-2Pl.I/td> You (pl.) have come to ask us questions. We also find examples of what seem to be 1Sg.III -ene (#9), 3Pl.I -ko (#6).
For Zerenkel, Ramat provides tac-ku "hit-3Sg.I", bar-kute "give-3MSg.II"; we've already seen -na "-1Sg.I", -ci "3FSg.I". Hunting through the untranslated texts might provide a fuller paradigm.
So it looks like this system is common across Mubic. What about its closest relatives, the Northern Guera languages? There, we find something reminiscent of Mubic, but a bit different: a distinction between preverbal subject proclitics (which do always seem to mark the subject, not the object) and postverbal subject suffixes, whose functions include marking subject continuity and object relativisation.
The best described of these, East Dangla (Shay 1999:125), has a distinction between preverbal subject markers (the default) and postverbal subject suffixes used when the subject is the same as in previous clauses. But the postverbal subject suffixes do not affect the interpretation of the immediate preverbal element, which remains a subject. Even if the object is fronted, the subject continues to intervene between it and the verb (ibid:130). Unfortunately, Shay does not describe the syntax of relative clauses in any detail, but every example of non-subject relativisation provided (pp. 95#24, 188#57, 213#63, 252#43) displays subject suffixes rather than subject prefixes.
For Bidiya, Alio (1986:222) makes the latter point explicit: object relativisation forces the verb of the relative clause to use the subject agreements suffixes, rather than the proclitics. The functional distinction between the two does not seem to be otherwise discussed, but examples (e.g. p. 347) again make it clear that the suffixes do not affect the interpretation of the preverbal position, which continues to be reserved for subjects.
For Migama, Jungraithmayr and Adams' (1992) sketch grammar reveals three different subject proclitic series, but no suffixed subject agreement series at all.
Further afield in the Southern Guera languages, we find even less in common with the Mubic system. As far as the materials available to me suggest, it seems that subjects there are simply preverbal irrespective of pronominal or morphological status, and irrespective of the syntactic context.
In Barain, "Subjects are primarily encoded by word order, being the only argument before the verb. Subjects pronouns appear to occur in the same syntactic position as other subjects" (Lovestrand 2012:132); pronominal elements suffixed to the verb are simply direct or indirect objects. The same structure applies in relative clauses (ibid:189). For Gali (Jungraithmayr and Peust 2019) and Saba (Jungraithmayr 2020), the brief sketches available likewise indicate that subject pronouns only precede the verb.
Putting this together, we start to get an idea of where the Mubi system might have come from. I assume that proto-Chadic basic word order was VSO, as still in some of its descendants. In proto-East Chadic B, the default word order was probably already SVO (fully generalised in Southern Guera), but VS was still preferred in object extraction contexts and for old information subjects, morphologising into a new subject suffix series in Northern Guera and Mubic. Old information subject contexts would notably have included object focus, a context for which fronting is cross-linguistically widely preferred; after the morphologisation, this would have led to the subject suffix series becoming obligatory when the object directly precedes the verb. Mubic overgeneralised this correlation, reinterpreting the subject suffix series as requiring the NP preceding the verb to be interpreted as its object.