Is engine braking bad for the car?
No. Engine braking is a normal, designed behaviour, and in most respects it is gentler on the car than using the brakes.
Engine braking is the slowing that happens when you lift off the accelerator in gear. The throttle closes, the engine can no longer draw air freely, and the pistons work against that restriction. The wheels are turning the engine rather than the engine turning the wheels, and that resistance slows the car.
The concerns people raise are largely unfounded.
It does not wear the engine. The loads involved are well within what the engine handles under normal driving, and the components are lubricated exactly as usual.
It does not use more fuel — it uses less. Modern fuel-injected engines cut fuel delivery entirely when coasting in gear above idle speed. This is called deceleration fuel cut-off, and it means engine braking in gear uses essentially no fuel at all, while coasting in neutral burns fuel to keep the engine idling. Descending a long hill in gear is measurably more economical than in neutral.
It saves brake wear, which is the practical argument for it. On a long descent, riding the brakes heats them until the fluid can boil and braking effectiveness fades. Engine braking carries much of that load without generating heat in the brakes.
What is genuinely bad practice is aggressive downshifting to slow quickly — dropping several gears at speed sends the engine toward its rev limit and puts real load on the clutch and drivetrain. Change down progressively.
In an automatic, selecting a lower gear or a dedicated braking mode achieves the same thing.
Electric and hybrid cars do something related but different: regenerative braking recovers energy rather than dissipating it.