The transmission fluid pressure is supposed to be proportional to the vehicle speed, and it has a sensor apparently to see if it does. It decides what gear to shift to based on the vehicle speed, which it gets from the fluid pressure. Something wrong w/all that, that’s what that transmission code probably means. If the transmission is shifting ok, behaving like it should, then the problem can probably be deferred, either the sensor is out of calibration, or the pressure isn’t being regulated accurately enough to meet the diagnostic system’s requirements, which could mean you just need a proper transmission service.
The other problem, the p0421, that’s the one you should focus on resolving, b/c CA emissions definitely won’t let you renew your registration with that one. The way it works, there’s two o2 sensors installed and monitored by the engine computer, one before the cat, and one after. If the cat is working, that effects the o2 content from the output of the cat, so the sensor outputs will be different. Your mechanic is saying the two are “mirroring” each other, which is another way of saying the pre cat o2 and the post cat o2 output are the same. No good, you are probably looking at needing a new cat.
Why the scanners are not consistent, hard to say. It’s like asking why the desk top computer you used yesterday, doesn’t work today. Hardware compatibility problems, software compatibility problems, power supply problems, etc. all could cause that sort of thing. Suggest to rely on what the Mazda specific scan tool says.