The DESeq2 vignette, under session "Group-specific condition effects, individuals nested within groups", suggests to use a trick from edgeR and "add a column ind.n which distinguishes the individuals nested within a group".

More specifically, to avoid modeling with ~ ind + grp*cnd, which raises a matrix rank issue, the vignette suggests to use ~ grp + grp:ind.n + grp:cnd, with ind.n being the added column from above.

My question is whether grp can be interpreted as the comparison between individuals within the different groups of grp when cnd is zero (or the reference value), as it would be the case for ~ ind + grp + cnd + grp:cnd.

Any help would be much appreciated.

The main effect is not interesting in this case, and should not be interpreted. We don't have random effects, so grp represents the group effect for one pair (the reference pair).