Why don't C++ compilers define operator== and operator!=?

The argument that if the compiler can provide a default copy constructor, it should be able to provide a similar default operator==() makes a certain amount of sense. I think that the reason for the decision not to provide a compiler-generated default for this operator can be guessed by what Stroustrup said about the default copy constructor in "The Design and Evolution of C++" (Section 11.4.1 - Control of Copying):

I personally consider it unfortunate that copy operations are defined by default and I prohibit copying of objects of many of my classes. However, C++ inherited its default assignment and copy constructors from C, and they are frequently used.

So instead of "why doesn't C++ have a default operator==()?", the question should have been "why does C++ have a default assignment and copy constructor?", with the answer being those items were included reluctantly by Stroustrup for backwards compatibility with C (probably the cause of most of C++'s warts, but also probably the primary reason for C++'s popularity).

For my own purposes, in my IDE the snippet I use for new classes contains declarations for a private assignment operator and copy constructor so that when I gen up a new class I get no default assignment and copy operations - I have to explicitly remove the declaration of those operations from the private: section if I want the compiler to be able to generate them for me.

Even in C++20, the compiler still won't implicitly generate operator== for you

struct foo
{
    std::string str;
    int n;
};

assert(foo{"Anton", 1} == foo{"Anton", 1}); // ill-formed

But you will gain the ability to explicitly default == since C++20:

struct foo
{
    std::string str;
    int n;

    // either member form
    bool operator==(foo const&) const = default;
    // ... or friend form
    friend bool operator==(foo const&, foo const&) = default;
};

Defaulting == does member-wise == (in the same way that the default copy constructor does member-wise copy construction). The new rules also provide the expected relationship between == and !=. For instance, with the declaration above, I can write both:

assert(foo{"Anton", 1} == foo{"Anton", 1}); // ok!
assert(foo{"Anton", 1} != foo{"Anton", 2}); // ok!

This specific feature (defaulting operator== and symmetry between == and !=) comes from one proposal that was part of the broader language feature that is operator<=>.