NB: you are at favourable, IMPs, long match.
2D was a multi (weak two in a major or strong in a minor)
3H was pass-or-correct
Partner's double of 3H is 'take-out', but it may transpire to be take-out of spades: you work it out from opener's next action.
Partner's second double is take-out of spades.
Partner could have passed over 3H and doubled 3S next round to show a lighter take-out double, but has a hand not prepared to risk 3H being passed out.
Does it make a difference if the auction is as it was in the other room, namely
2S P 3S Dble
P ?

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2♦ P 3♥ x
3♠ P P x
P ?