Do I use Synthetic oil on Honda Odyssey 2006?

No, not unless you want to. The Honda dealer just uses a "bulk generic" oil. They don't even use Honda oil. Whatever they can purchase in 55 gallon drums for as cheap as they can purchase it, is what they buy.

Do use an original Honda filter though. It has an anti-drainback valve in it which keeps more oil in the motor to help prevent all the oil from going back into the oil pan when the motor is not running.