Any problem with installing two identical circuit breakers in series?

Logically, this makes some sense.

Practically, there is no breaker panel configured to let you do this. You cannot modify the panel, physically, without invalidating the U/L certification.

Actually the system availability goes down for mechanisms in series, because you multiply the individual available availabilities.

There is no problem here.

In fact, this is often how subpanels are configured; the main panel has a breaker that feeds the subpanel, and the subpanel has individual circuits protected with their own breakers. The subpanel will have an amp rating. The minimum of the subpanel rating and the ampacity rating of the wire feeding the subpanel will dictate the appropriate breaker for the main panel.

Appliances that require their own local disconnects (such as air conditioners, water heaters, etc.) frequently use a small enclosure with space for just one circuit breaker. This could be a good option for whatever application you are thinking of.

Two breakers in series would reduce the risk of one failing in a way that stays in a closed (conducting) position. But modern breakers today have an extreme low risk of this happening. If I were that paranoid, I'd put a breaker and a fuse in series. But I'm not.

Putting two breakers of same size and rating in series can be dangerous. manufacturers give the short circuit breaking current for ONE device. Modern CB operated with the magnetic force generated during the short circuit. This magnetic repulsion of the contacts is linked to the energy generated during the short circuit. The breakers are designed to open according to this energy generated during the short circuit. the opening time and speed is defined according to this energy. If two breakers are in series it impossible to predicte the energy at the level of the two devices. The energy is distributed over the two devices. And the operation time and openning speed can be slower than with only one breaker. Putting two breaker of same size and same rating can be dangerous.

Most generator manual transfer switch-over panels require you to do just this. The wire coming out of the breaker and going to a load (normally) is disconnected and channeled over to a second panel with another breaker and then then the biter-end coming out of the second panel is tied back to the load. In short, two breakers in series.

I have to agree: Connecting two circuit breakers of the same frame size and trip rating in series could be dangerous, depending on the application. When two circuit breakers of the same trip rating are connected in series it is impossible to predict which circuit breaker will trip first on a fault condition or on overload. The proper operation of the circuit breakers and clearing time would be affected. Connecting two circuit breakers of the same frame size and trip rating is simply not a good practice.

"Suppose I want to reduce risk of a circuit breaker malfunctioning and failing to disconnect a shorted circuit."

If you have a good quality panel... Square D, Eaton, Siemens, etc.. you really have no reason to worry about a circuit breaker malfunctioning in such a way that it doesn't trip if your load is greater than its ampacity. I have never seen this happen to any manufacturers circuit breakers not named Zinsco. In short, circuit breakers work and they work well you dont have to be afraid of them failing. If you are really nervous, install a arc fault circuit interrupter breaker in the existing ones place... any arcing and it will trip. Or an arc fault/ GFI combo breaker that will protect you from every angle.