Why is shared_ptr<void> legal, while unique_ptr<void> is ill-formed?

It is because std::shared_ptr implements type-erasure, while std::unique_ptr does not.

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Since std::shared_ptr implements type-erasure, it also supports another interesting property, viz. it does not need the type of the deleter as template type argument to the class template. Look at their declarations:

template<class T,class Deleter = std::default_delete<T> > 
class unique_ptr;

which has Deleter as type parameter, while

template<class T> 
class shared_ptr;

does not have it.

So, why does shared_ptr implement type-erasure?

Well, it does so, because it has to support reference-counting, and to support this, it has to allocate memory from heap and since it has to allocate memory anyway, it goes one step further and implements type-erasure — which needs heap allocation too. So basically it is just being opportunist!

Because of type-erasure, std::shared_ptr is able to support two things:

• It can store objects of any type as void*, yet it is still able to delete the objects on destruction properly by correctly invoking their destructor.
• The type of deleter is not passed as type argument to the class template, which means a little bit freedom without compromising type-safety.

Alright. That is all about how std::shared_ptr works.

Now the question is, can std::unique_ptr store objects as void*? Well, the answer is, yes — provided you pass a suitable deleter as argument. Here is one such demonstration:

int main()
{
    auto deleter = [](void const * data ) {
        int const * p = static_cast<int const*>(data);
        std::cout << *p << " located at " << p <<  " is being deleted";
        delete p;
    };
    
    std::unique_ptr<void, decltype(deleter)> p(new int(959), deleter);
    
} //p will be deleted here, both p ;-)

Output (online demo):

959 located at 0x18aec20 is being deleted

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You asked a very interesting question in the comment:

In my case I will need a type erasing deleter, but it seems possible as well (at the cost of some heap allocation). Basically, does this mean there is actually a niche spot for a 3rd type of smart pointer: an exclusive ownership smart pointer with type erasure.

to which @Steve Jessop suggested the following solution,

I've never actually tried this, but maybe you could achieve that by using an appropriate std::function as the deleter type with unique_ptr? Supposing that actually works then you're done, exclusive ownership and a type-erased deleter.

Following this suggestion, I implemented this (though it does not make use of std::function as it does not seem necessary):

using unique_void_ptr = std::unique_ptr<void, void(*)(void const*)>;

template<typename T>
auto unique_void(T * ptr) -> unique_void_ptr
{
    return unique_void_ptr(ptr, [](void const * data) {
         T const * p = static_cast<T const*>(data);
         std::cout << "{" << *p << "} located at [" << p <<  "] is being deleted.\n";
         delete p;
    });
}

int main()
{
    auto p1 = unique_void(new int(959));
    auto p2 = unique_void(new double(595.5));
    auto p3 = unique_void(new std::string("Hello World"));
}  

Output (online demo):

{Hello World} located at [0x2364c60] is being deleted.
{595.5} located at [0x2364c40] is being deleted.
{959} located at [0x2364c20] is being deleted.

Hope that helps.