Can a Functor / Applicative be tied to one specific type or structure?

Yes, your understanding is largely correct. In particular, any specific Applicative, say one named Foo, has an associated specialization of the function pure with type signature:

pure :: a -> Foo a

that must work for any type a selected by the caller, such as:

> pure 10 :: Foo Int
> pure length :: Foo (String -> Int)

So, whatever Foo is, it has to be able to "handle" any provided type without limitations because pure can technically be applied to any type without limitations.

One cautionary note, though. The idea that a functor f "wraps" data, so that f Int is somehow a "container" of Int values, can be a helpful intuition and is often literally correct (e.g., lists, trees, etc.), but it's not always strictly true. (Some counterexamples include the functors IO, (->) r, and Const b, which "contain" values in a very different sense than real containers.)

For "regular" Functors and Applicatives, you're right; they need to be able to handle values of any type. This is known as parametric polymorphism. If you have a type that you think is almost a Functor except that it can't do that, then consider the MonoFunctor typeclass from the mono-traversable package. It's the same idea as Functor, except with a single valid element type baked in. I'm not aware of any packages that have a monomorphic equivalent to Applicative. I think this is because <*> uses values of 3 different types inside the same container, so it doesn't have a good monomorphic analogue.