Try:
$Version
(* "12.3.0 for Microsoft Windows (64-bit) (May 10, 2021)"*)
Solve is symbolic solver need exact values:
Solve[{m1 + m2 == m3, v1 + v2 == v3, m1/v1 == SG1, m2/v2 == SG2,
m3/v3 == SG3, m1/((m1 + m2)) == Cw, SG1 == 135/100, SG2 == 1,
Cw == 52/100}, SG3, MaxExtraConditions -> All]
(*{{SG3 -> ConditionalExpression[675/584,
And[-13 + 25 Cw == 0, (-12) m1 + 13 m2 == 0, (-25)
m1 + 13 m3 == 0, -27 + 20 SG1 == 0, -1 + SG2 == 0, (-20)
m1 + 27 v1 == 0, (-12) m1 + 13 v2 == 0, (-584)
m1 + 351 v3 == 0]]}}*)
With numeric values gives errors,but can solve:
Solve[{m1 + m2 == m3, v1 + v2 == v3, m1/v1 == SG1, m2/v2 == SG2,
m3/v3 == SG3, m1/((m1 + m2)) == Cw, SG1 == 1.35, SG2 == 1,
Cw == 0.52}, SG3, MaxExtraConditions -> 8]