Area moment of inertia and mass moment of inertia are two different things.
A) Mass moment of inertia (or moment of inertia):
This is the resistance offered by a solid body when subjected to rotation (or application of torque). The mass moment of inertia is given by
$$I = \int r^2 dM $$
The radius of gyration is the distance at which the mass can be considered to be concentrated (with respect to the axis of rotation). For two wheels with the same overall mass but different distribution of the mass, the one with mass concentrated near the axis of rotation has less moment of inertia. This is the 'rotational' equivalent of the 'mass' in translational motion, thus sometime also known as the 'angular mass'. Moment of inertia has units $kg m^2$. Here is a list of moments of inertia some regular geometric shapes. Check out this video by the YouTube channel 'smarter every day' for a physical sense of the moment of inertia.
B) Area moment of inertia (or second moment of area):
The area moment of inertia is a property of a $2$D section or surface of a body. For a solid body, the area moment of inertia can be defined only for the cross sections. The second moment of area is not to be confused with mass moment of inertia. The area moment of inertia is used to characterize the resistance offered by a body (such as a beam) for deflection under load. Its units are $m^4$.
Here is a list of area moments of inertia of a few regular $2$D shapes. Note that, for a circular shaft, the polar moment of inertia is more relevant.
Coming to the question,
So my question is whether this $r_{mass}$ and $r_{area}$ are same for
any arbitrary object? Of course the axes about which integral is
evaluated are the same.
Provided that a solid body has uniform cross sectional area and uniform second moment of area (both about the axis of rotation), $r_{mass}$ and $r_{area}$ will be same if the material is isotropic (i.e. uniform density).
Proof:
For an object with uniform thickness $h$ unit and constant cross sectional area $dA$ and constant second moment of area $I_{area}$ along the thickness, the total mass will be
$$ M = \int dM = \rho h \int dA $$.
In such as case,
$$r_{mass} = \sqrt{\frac{\int r^2 dM}{\int dM}} = \sqrt{\frac{\rho h \int r^2 dA}{\rho h \int dA}} $$
where, $\rho$ is the constant density of the material. This is same as $r_{area}$. However, it is to be noted that the mass moment of inertia was found out for the entire body whereas the area moment of inertia is only valid for a surface of the body (cross section) thus it needs to be same along the axis, same holds true for area of the cross-section itself.
