C++: Passing object to class constructor, how is it stored?

Yes, the data member X will be copy-initialized from the constructor parameter X.

If you declare the data member X as reference, then no copy operation happens. E.g.

class Foo {
private:
    std::vector<double>& X;
public:
    ~Foo() = default;
    Foo(std::vector<double>&);
};

Foo::Foo(std::vector<double>& X):
        X(X)
{}

Then

std::vector<double> v;
Foo f(v); // no copies; f.X refers to v

Rest assured. The member's type is vector<double>, so the compiler will look for a constructor overload in the vector<double> class that matches the provided argument type (vector<double>&).

The best match it will find is the const vector<double>& overload - the copy constructor.