No, there is no need for the Rosenberg noncommutative scheme that the categories be abelian in general; whatever being said in his 1998 paper he does not mean so. He defines a relative scheme over a base category given by exactness properties of direct and inverse image functors describing covers used for gluing and the affinity property. In general one has to be careful, with formulating correctly the exactness properties for nonabelian context.

Of course, to justify this one needs to say that the Durov's schemes are determined by the categories of quasicoherent modules. I do not know if the reconstruction theorem a la Gabriel-Rosenberg holds for the generalizes schemes of Durov. Durov glues schems along categorical localizations which he calls for some reason "pseudolocalizations". They just have the correct exactness properties. On the other hand, for commutative monads, Durov has two nice versions of prime spectra; now one should compare those with a version of Rosenberg's spectrum for nonabelian context. Now one version if the spectrum for right exact categories of Rosenberg. What is a right exact structure for the case of categories like the ones in Durov's work ? Well Durov has spent some time to develop a theory of vectoids which generalize topoi, but also the categories of modules over finitary monads in Set and, more generally, the categories of quasicoherent sheaves of $\mathcal{O}$-modules over generalized schemes. Durov wrote a draft text in Russian about vectoids (which I have seen but is not released yet), and a video of a talk at Steklov. Is there a canonical right exact structure on a vectoid for which the spectrum of the right exact category in the sense of Rosenberg gives a sensible reconstruction theorem, it would be very interesting to investigate.

Remark: I believe that that the quantum flag variety of Lunts-Rosenberg is isomorphic in the case of $SL_n$ to the flag variety studied in my thesis from the dual point of view and with explicit Ore localizations. I never had time to check and publish all the details.