You think that you see the image in the mirror because the real image is formed at a position which devoid of any objects close to it which can be used for reference. It is difficult to get the eye to focus on a real image formed in "thin air".
The only other object which you can focus on is the mirror and so you see both the real image and the mirror but cannot ascertain there relative separation.
Get the mirror to form an image of a distant object as in the left hand diagram.
Where I have the label pin place a bit of tracing paper (a screen which allows some light through) and move it towards and away from the mirror as shown in the left diagram until a sharp image of the distant object if formed.
You will note that the image is not formed "in the mirror" but in front of the mirror although the mirror can also be seen.
You can identify the position of the real image formed by the mirror by using the method of no-parallax.
Pin a pin or small object between the mirror and the eye and you should see the pin and the distant object.
Move your eye left to right as per the right-hand diagram and adjust the position of the pin until there is no relative movement between the pin and the image of the distant object - a position of no parallax.
The pin is now located in the same region as the image of the distant object.
Here you will find the same method described for the real image formed by a converging lens.