In the original case, the printer is behaving as designed. Most customers don't move cartridges from one printer to the next. If a well-used cartridge, that has already reached the 'low' level, is moved to a different printer, HP makes the assumption that this cartridge has probably been refilled, and tags it as 'non HP.' You can print normally with this cartridge but some product features may be disabled.
In the case of a new printer with only a few hundred pages displaying 'non-HP supplies installed,' there is a defective component in the system. This customer should call HP support and get the printer fixed under warranty.
I hope this helps,
Alan_R