Substitution rule for sums of similar objects?

Try the following:

obj /. HoldPattern@Plus@a : (h : Subscript)[A, _] .. :> h[A, ## & @@ Last /@ {a}]

It can be even shorter:

obj /. HoldPattern@+a : (h : Subscript)[A, _] .. :> h[A, ## & @@ Last /@ {a}]

Here's an alternative that will work with subscripted As that have more than one subscript, which may (or may not) be what you want:

obj /. HoldPattern[Plus[seq : Subscript[A, __] ..]] :>
  Subscript[A, Sequence @@ Cases[{seq}, Subscript[A, inds__] :> inds]]

Block[{Plus = Subscript[#[[1]], ## & @@ (#2 & @@@ {##})] &},  obj] // TeXForm

$\LARGE\frac{A_{2,3} A_{3,4,5,6} A_{1,2,3,4,5}}{A_{1,2} A_{4,5,6} A_{3,4,5,6,7,8}}$

Alternatively,

obj /. Plus -> (Subscript[#[[1]], ##&@@(#2&@@@{##})]&) // TeXForm

$\LARGE\frac{A_{2,3} A_{3,4,5,6} A_{1,2,3,4,5}}{A_{1,2} A_{4,5,6} A_{3,4,5,6,7,8}}$