Arguing that the 10th Fleet only had a single assignment is possible, but not all that consistent with what we know of the assignments of all the other described fleets. The 3rd was considered for participation in liberating DS9, even if it had commitments at Earth, so it wasn't flat out impossible for it to have varied assignments. The 2nd and the 5th that eventually were included in this liberation were explicitly said to be pulled back from other, distant assignments.
All the back-and-forth rushing of fleets we saw makes it look unlikely that fleet identity had anything to do with specific assignments or locations. For all we know, the ten (or slightly more) fleets were created for the purposes of the Dominion War only, all
of them being mixtures of UFP and ally assets, and some other war would have seen different forces differently arranged.