I use after marketfluids in my Hondas. I have for quite some time, and I haven’t noticed any problems.
My Civic has aftermarket power steering fluid (Gunk’s “Power Steering Fluid for Honda Vehicles”), aftermarket brake/clutch fluid (DOT3), aftermarket coolant (Prestone), aftermarket manual transmission oil (synthetic 10W-30 oil), and aftermarket motor oil (5W-30, usually Prestone or Valvoline).
The only Honda brand fluid I currently use is gear oil in the differential on a motorcycle that has a shaft drive instead of a chain, and only because I don’t know of a readily available suitable aftermarket substitute.