From the Super Mario Wiki
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'Stache or Stache (Horn for Bowser) is a a statistic in the Mario & Luigi RPG series, similar to Luck in other RPGs. It usually gives Mario and Luigi or Bowser better chances at getting a "Lucky" hit in battle. in Mario & Luigi: Superstar Saga and Mario & Luigi: Partners in Time it gives discounts to items bought at shops, and sells items at higher prices. Baby Mario and Baby Luigi also have this stat in Mario & Luigi: Partners in Time, despite not having mustaches. In Mario & Luigi: Bowser's Inside Story it affects the type of scratch card the player gets after purchasing items. Since none of these features are present in Mario & Luigi: Dream Team the stat is only used for lucky hits and nothing else while Mario & Luigi: Paper Jam it also increases the chances of enemies dropping items after battle.
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