Does Pokemon Go take into account STAB in gym battles?

Yes, it does. But only at 1.25x. Here is the decoded protobuf file. In it is this:

Items {
  TemplateId: "BATTLE_SETTINGS"
  BattleSettings {
    RetargetSeconds: 0.5
    EnemyAttackInterval: 1.5
    AttackServerInterval: 5
    RoundDurationSeconds: 99
    BonusTimePerAllySeconds: 10
    MaximumAttackersPerBattle: 20
    SameTypeAttackBonusMultiplier: 1.25
    MaximumEnergy: 100
    EnergyDeltaPerHealthLost: 0.5
    DodgeDurationMs: 500
    MinimumPlayerLevel: 5
    SwapDurationMs: 1000
  }
}

Since there has been posted proof now, the STAB modifier is indeed in the game, just not as high as it used to be in the other games.

The exact calculations are

• super effective: x1.25
• not very effective: x0.75
• STAB: x1.25