Yes I alsohave this question, or a similar one. There are a FEW times I can imagine saying this, but not nearly as much as I can imagine saying 'They seem to HAVE the same system.' - For example 'In front wheel drive cars they normally use this system. This car is all-wheel drive but it seems to have the same system.'
'Apply' a system might be used where a very complicated set of rules were laid down in a given situation, maybe in Test Cricket or something, 'When the lighting is bad, the umpires apply the Duckworth-Lewis system. They apply the same system when it rains or when anything stops play.' Random example, but that's just my point, it's pretty hard to find a use for 'applying a system' otherwise.
I wonder if there are other words DuoLingo will allow instead of the word "apply." I have never heard anyone "apply" a system. It would have to be "use," "utilize," or "have." You can't really apply a system because the fact that you have a system in place means it is automatically applied all the time.