Template class with template container

You should use template template parameters:

template<typename T, template <typename, typename> class Container>
//                   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
class MyMultibyteString
{
    Container<T, std::allocator<T>> buffer;
    // ...
};

This would allow you to write:

MyMultibyteString<int, std::vector> mbs;

Here is a compiling live example. An alternative way of writing the above could be:

template<typename T,
    template <typename, typename = std::allocator<T>> class Container>
//  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
class MyMultibyteString
{
    Container<T> buffer; // <== No more need to specify the second argument here
    // ...
};

And here is the corresponding live example.

The only thing you have to pay attention to is that the number and type of arguments in the template template parameter declaration must match exactly the number and type of arguments in the definition of the corresponding class template you want to pass as a template argument, regardless of the fact that some of those parameters may have default values.

For instance, the class template std::vector accepts two template parameters (the element type and the allocator type), although the second one has the default value std::allocator<T>. Because of this, you could not write:

template<typename T, template <typename> class Container>
//                             ^^^^^^^^
//                             Notice: just one template parameter declared!
class MyMultibyteString
{
    Container<T> buffer;
    // ...
};

// ...

MyMultibyteString<int, std::vector> mbs; // ERROR!
//                     ^^^^^^^^^^^
//                     The std::vector class template accepts *two*
//                     template parameters (even though the second
//                     one has a default argument)

This means that you won’t be able to write one single class template that can accept both std::set and std::vector as a template template parameter, because unlike std::vector, the std::set class template accepts three template parameters.