Summary: for creatures who fail both saves, both effects stay in place until broken or the duration runs out. A target will get 1-2 saving throw attempts to break the effect at the end of their turn. See below.Basic Rules, Appendix A, Conditions (PHB 290)
If multiple effects impose the same condition on a creature, each instance of the condition has its own duration, but the condition's effects don't get worse. A creature either has a condition or doesn't.
Round 1: Pipes impose "frightened" condition. Duration 1 minute. Unlike the fear spell, there is no special requirement to attempt another save to break the spell.
Frightened is defined as having disadvantage on skills/attacks, and unable to move closer to source of fear.
Round 2: Fear spell imposes "frightened condition." Duration 1 minute. While the effect can't get worse, fear specifically says the targets are unable to attempt another save unless out of sight of fear source.
The Fear spell imposes non-condition penalties, including drop items and run away.
Foes will save twice each round, assuming the fear prerequisite is made, because each instance has its own duration. If a foe saves against the Pipes, they still have to get out of sight to break the fear spell. If a foe only saves against the fear spell, they don't have to run away anymore, but they still have to save against the Pipes to lose the frightened condition. It doesn't go away just because they made one save. Only the fear spell stopped working.
Now, if two folks had Pipes of Haunting, the same would apply even if the effects were identical, and a save against one doesn't make one immune to the Pipes of another.
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