As long as the oil has the correct weight (75w-90) and is SAE approved, there should not be damage to the manual transmission.
The chemistry affects the flow characteristics with age and temperature.
Fully synthetic oils maintain their viscosity over a wider temperature and for a longer period of time and is recommended when cost is not a concern.
For cost concerns, semi-synthetics and conventional can be used, but require inspections over a "shorter" period of time/mileage than fully synthetic oils.
This answer will change drastically if it was not a manual transmission (instead, say an automatic transmission where the fluid not only provides lubrication but affects the hydraulic flow characteristics of the transmission completely. Honda automatic transmissions are well known to have this requirement of working only with OEM fluid)