Heat olive oil in a skillet over medium heat. Add onion; cook and stir until translucent, about 5 minutes.
Combine the onion, mangoes, peach, honey, habanero peppers, and mustard in a food processor. Process until smooth; mix in salt, paprika, white pepper, cumin, and allspice. Add brown sugar; continue to process until well combined.
Pour mango-habanero mixture into a large saucepan. Bring to a boil. Add white vinegar and apple cider vinegar, stirring frequently until thoroughly combined. Boil until thickened, 10 to 12 minutes. Pour sauce into jars or containers.