> Without syntax to say "this is an object",

How about this syntax?

    #include "oo.h"

    DEFINE_CLASS(Counter,
     int METHOD(Counter,getValue,NO_ARGS);,
     void METHOD(Counter,increment,NO_ARGS);,
     void METHOD(Counter,decrement,NO_ARGS);,
     void METHOD(Counter,multiplyThenAdd,(int multiplyBy, int add));
    );

That's valid C, given appropriate preprocessor macro definitions. It is a bit ugly, due to the limitations of the C preprocessor. (Maybe someone else can do better than me.) If C had a more powerful preprocessor, but was otherwise unchanged, the syntax could be a lot nicer.

The DEFINE_CLASS() macro expands to declare a function:

    Counter* newCounter(void);

Why isn't the return value of that function an object?

Here is the contents of "oo.h":

    #include "cloak.h"
    
    #define JOIN(...) JOIN_IMPL(__VA_ARGS__,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,)
    #define JOIN_IMPL(_1,_2,_3,_4,_5,_6,_7,_8,_9,_10,_11,_12,_13,_14,_15,_16,_17,_18,_19,_20,_21,_22,_23,_24,_25,_26,_27,_28,_29,_30,_31,_32,_33,_34,_35,_36,_37,_38,_39,_40,_41,_42,_43,_44,_45,_46,_47,_48,_49,_50,_51,_52,_53,_54,_55,_56,_57,_58,_59,_60,_61,_62,_63,...) \
     _1 _2 _3 _4 _5 _6 _7 _8 _9 _10 _11 _12 _13 _14 _15 _16 _17 _18 _19 _20 _21 _22 _23 _24 _25 _26 _27 _28 _29 _30 _31 _32 _33 _34 _35 _36 _37 _38 _39 _40 _41 _42 _43 _44 _45 _46 _47 _48 _49 _50 _51 _52 _53 _54 _55 _56 _57 _58 _59 _60 _61 _62 _63 
    
    #define DEFINE_CLASS(name,...) \
     typedef struct name name; \
     typedef struct name { \
      void *_privateData; \
      JOIN(__VA_ARGS__) \
      void (*destroy)(name**); \
            } name; \
     name* new##name(void)
    #define METHOD(class,method,args) \
     (*method)(class* COMMA_IF(IS_PAREN(args)) WHEN(IS_PAREN(args))(EXPAND(EXPAND args)))
    #define NO_ARGS

Where cloak.h is https://github.com/pfultz2/Cloak/blob/master/cloak.h

The above macros are admitttedly very hairy. If C had a better preprocessor, but was otherwise unchanged, they could look a lot nicer.

> I never said such a thing. Would appreciate if you didn't put words in my mouth.

A normal part of dialogue is, "I'm going to repeat your point back to you in my own words, and you can either agree with my restating of it, or point out at which point I've misunderstood you". That's what I was doing. "Would appreciate if you didn't put words in my mouth" is unnecessary hostility.

> In my concrete example, given that FILE and DIR were classified as objects, to which of the so-called objects does F(const char*, FILE, DIR) belong?

Take any language which you agree is "OO". Add one new feature (if it isn't already there): functions/methods which don't belong to any class/object. Now the function you are talking about is possible in that language. Did the language thereby suddenly cease to be OO when we added that feature? Most people would disagree with "Yes". But if "No", what is the actual difference between C and that language?