Honda officially recommends 10W-30 API Service Category SJ petroleum oil for use in all* engines because petroleum oil is what is used when the engine is certified by the EPA and California ARB rules.
Synthetic oil is acceptable, provided it is 10W-30 and API Service Category SJ.
As others have said, it's not so much about the oil, but that you change it per the maintenance schedule. Often, it's not the lubrication or oil itself that fails, but the additives and detergents that can only absorb so much carbon and other byproducts produced by internal combustion engines.
* The exception for this is snowblowers, which should use 5W-30.
Caveat: I work for Honda, but the preceding is my opinion alone.