To be fair, the owner's manuals don't specify synthetic oil. They don't even recommend it. All that's necessary for the warranty to be honoured is that the specs of the oil used match with what is specified in the manual. The OCIs identified in the Maintenance Schedule are based on the recommended oil specs, after all.
All that said, synthetic oil is a much better option for a turbocharged engine, and the added cost of synthetic is well worth the long term benefits the engine gets from it. If you have access to and can afford synthetic, you should use it.
Here is the excerpt from the manual:
▼ Recommended Oil U.S.A., Canada SKYACTIV-G 2.0, SKYACTIV-G 2.5 Use SAE 0W-20 engine oil. Mazda Genuine Oil is used in your Mazda vehicle. Mazda Genuine 0W-20 Oil is required to achieve optimum fuel economy
SKYACTIV-G 2.5T Use SAE 5W-30 engine oil. Mazda Genuine Oil is used in your Mazda vehicle. Mazda Genuine 5W-30 Oil is required to achieve optimum fuel economy
Outside US, Canada the recommendation in the manual is to use SAW 5W-30. That's all it says.