I don't think Mobil 0W-20 would be a good choice for transmission oil...
That oil will not have hardly any anti-wear additives like transmission fluid does. The ad-pack is totally different. The viscosity will be wonky too. Multi-grade oils were made with the assumption they'd be used in a specific application. The OE Dex3 fluid will behave quite differently than a multi-grade 0W-20 which is going to behave like a 0 grade oil when cold (way too thin). The synchros and bearings were designed with Dex3 ATF fluid properties in mind, and anything else will deliver abnormal performance. How do I know? BTDT experience...
If you wanted, they do make full synthetic base stock Dex3 ATF. Valvoline Maxlife comes to mind. You could also try Dex6 since it is SUPPOSED to be backward compatible with Dex3 specifications.
Honda transmissions used to call for 10W-30 motor oil at one point in time. The difference is that they were designed to use that from the start, and the ZDDP content of oil was higher back then adding to the oil's usefulness in extreme pressure scenarios.