Toyota and Honda have each other as direct competitors. If one of them sacrifices reliability, it's over. There are also second tier ones like Nissan making interchangeable cars, so if say Toyota stoops to Nissan's level, then they have to start selling at Nissan prices.
I would also say Ford, but their reputation for reliability applies only to their trucks.
Funny that Honda's hybrids seem to be licensing Toyota's eCVT magic, so it's quite a bit like Apple and Samsung, where they would be filling each other's pockets while also being at each other's throat.
When Ford and Mazda were working together they had some pretty reliable cars. My Focus outlasted two Hondas that I owned. The Hondas were "nicer" cars in terms of features and interior but mechanically the Ford kicked their asses.