[solved] understanding const variables

v3xX:
at the end of MyConstantClass.h

That's where you DECLARE it, but you never DEFINE it. See: What is the difference between a definition and a declaration in C++?

Also, you need to tell the compiler that your class's methods won't modify the internal state of the class:

#ifndef H_CONSTANTCLASS
#define H_CONSTANTCLASS
#include <Arduino.h>
class MyConstantClass {

  public:
    const byte CONT = 'b';
    
    void println(const char* msg) const {
      Serial.print("OK here: "); Serial.println(msg);
    }

    void println() const {
      Serial.print("Test: "); Serial.println(CONT);
    }
};

extern const MyConstantClass MCC;

#endif