Honda doesn't specifically say what the issue is with a different ATF. The viscosity is one thing, and for the most part differences in hydraulic fluid can be tolerated. However, one claim I've heard is that the fluid has to be formulated to be compatible with Honda's seals.
Quote:
http://www.odyclub.com/forums/52-2005-2010-odyssey/190921-honda-power-steering-fluid-equivalent.html
The problem with with Honda PSF is that there truly is nothing magical to it, as you said. It is the most narrowly specified hydraulic fluid I've ever seen. It does only one thing well, and that is, act as a power transmission medium in rotary vane pumps for a single motion system like a steering rack. That's it. However, it's an excellent specialty fluid in this app. It would be lousy in a recirculating ball system of any sort, whether it be one used in a steering rack, manlift, or any other application...but Dexron III can excel in these applications.
Current iterations of Dexron III have changed markedly, with compounds that will intercalate into modern synthetics used in transmission seals while remaining friendly to older spec transmission seals, and both classes of friction modifiers are now heavily halogenated. They're not even close to what they used to have in them 20 years ago. The rubber in our PSF pump is probably the same ultra-generic nitrile makeup used in a 1980's Honda (read this as very inexpensive). It doesn't need to be anything better or pricier.
Reason I know this? I've had this academic discussion before with friends of mine (and one relative) working for Exxon Mobil and a college buddy-now-materials scientist as an after-action discussion for the following event:
Two neighbors saw me doing a PSF change on my 2003 years ago, so they did a PSF change on their Odysseys as well. They didn't have enough Honda PSF on hand when finished...so they shared a bottle of ATF and filled up their PS reservoirs with Dex III. A couple months later, something didn't "feel right" in the steering, so they said to me.
Being that they did the PSF changeout based on my recommendation, I felt compelled to help them. I was shocked at how black the PSF was compared to mine, and theirs was newer! After draining the reservoirs, I assisted in removal of the pumps and their disassembly. Every single seal in either pump, to include the backplate O-ring which is a captive, non-moving item, had physically degraded. Rather than clean the pieces out and rebuild the pumps on both Gen 2's, they purchased new pumps and I helped with the installation, system cleanup (reservoir removal and flushing) and flushing the systems repeatedly with quarts of Honda PSF. I did not know at that point that they'd used a full reservoir's worth of Dex III ATF.