Use both values of $\pm$ in equations

The following function performs the transformation I've suggested in the comments:

Options[PMReduce] = {"SynchronizePM" -> False};
PMReduce[eqns_, vars_, dom_: Complexes, OptionsPattern[]] := Module[
  {
   pEqns,
   params,
   i = 0,
   step = Boole[! OptionValue@"SynchronizePM"]
   }, {pEqns, params} = Reap[
    Flatten@{eqns} //. {
      (a_ ± b_) :> (a + Sow[C[i += step]] b),
      (a_ ∓ b_) :> (a - Sow[C[i += step]] b)
      }
    ];
  Reduce[
   Append[
    pEqns,
    And @@ (# == 1 || # == -1 & /@ First@params)
    ],
   vars,
   dom
   ]
  ]

The "SynchronizePM" option controls whether the ± should be changed synchronously (i.e. a±b±c to a+b+c||a-b-c)

Examples:

PMReduce[Abs[x ± 2] ± 3 == 0, x, Reals]
(* (C[1] == -1 && C[2] == -1 && x == -1)
|| (C[1] == -1 && C[2] == -1 && x == 5)
|| (C[1] == -1 && C[2] == 1 && x == -5)
|| (C[1] == -1 && C[2] == 1 && x == 1) *)

PMReduce[Abs[x ± 2] ± 3 == 0, x, Reals, "SynchronizePM" -> True]
(* (C[0] == -1 && x == -1) 
|| (C[0] == -1 && x == 5) *)

I may need to give this more thought but it seems to me you actually want Or.

For example:

a_: 0 ± n_ := a - n || a + n

Solve[x ± 2 == 0, x]

Solve[Abs[x ± 2] ± 3 == 0, x]

{{x -> -2}, {x -> 2}}

{{x -> -5}, {x -> -1}, {x -> 1}, {x -> 5}}

---

In light of your edit to include Plot we could either allow the expansion above and modify Plot to operate on Or, or we could modify both Solve and Plot (etc.) to specially handle ±. Doing my best to read between the lines I think you prefer the latter, so I'll illustrate that.

ClearAll[PlusMinus]

Plot;
Unprotect[Plot];
PrependTo[DownValues[Plot], 
  HoldPattern[Plot[expr_, arg___]] /; ! TrueQ[$plotPlusMinus] :>
    Block[{$plotPlusMinus = True}, 
      Plot[#, arg] &[expr //. a_: 0 ± n_ :> {a - n, a + n}]
    ]
];
Protect[Plot];

Plot[x + Sin[±x] ± 1/2, {x, 0, 8}]

Notes

• This assumes the plot function is composed of Listable functions; if that does not hold manual threading can be added, but I first want to know if this is moving in a direction that pleases you or not.

• I did not localize the Plot variable(s). That can be done but again I first want to know if this is going to be useful.

Little improving magnificent code from user Lukas Lang:

SOLVE[eqns_, vars_, dom_: Complexes] := {{ToRules[Module[{pEqns, params, i = 0}, {pEqns, params} = 
Reap[Flatten@{eqns} //. (a_ \[PlusMinus] b_) :> (a + Sow[f[i++]] b)];
Reduce[Append[pEqns, And @@ (# == 1 || # == -1 & /@ First@params)], vars, dom]]]}[[All, -1]]}


SOLVE[x ± 2 == 0, x]
(* {{x -> 2, x -> -2}} *)

SOLVE[Abs[x ± 2] ± 3 == 0, x, Reals]
(* {{x -> -1, x -> 5, x -> -5, x -> 1}} *)