Brand new power steering pump whining?

Brand new power steering pump whining? - 1

Added (1). 2010 Honda Odyssey 90K miles, new pump, fluid, reservoir purchased from Honda

Added (2). Is this common to Hondas, like the famous clunking in the suspension?

It's probably a poor quality rebuild, take it back and try another but give this one a couple days for the air bubbles to diminish, it may quiet down.
Re: update - if it's a "new" Honda pump it just needs a little time for the air to boil out.

Some Honda's had a common problem with an O-ring that failed on the return line and it caused a small leak but also noise in the pump as air was drawn in but I believe it was mostly on Accords, could be the Ody too though if you have an O-ringed fitting on the intake.

You have air in the pump. Start the engine and turn the steering wheel fully right and then fully left at least 10 times. This will remove any air trapped in the system. If it is still whining then you bought a defective pump.

It is very important to bleed the air from the pump, when it is new, and just installed. If you start the engine with air in the pump, it will be noisy for life. To bleed the air from a new pump; install the pump, attach the belt, insure the pump is full of the proper fluid, jack up the front end with the tires off the ground. With the engine off, turn the steering wheel, stop to stop, 10 times. Check the level of fluid. Turn the wheel 5 more times, stop to stop. Start the engine. If you had the pump installed some where, take it back to the repair facility, and complain to the manager. If you installed it your self, you are out of luck.