There are two schools of angular velocity. One school could be called the inertial frame school, or Goldstein's school. In this school one talks about the angular velocity of a rigid body, as seen in the inertial space frame. In this school one uses the phrase "the angular velocity in the body frame" to mean the angular velocity as measured in the inertial frame but expressed in terms of basis vectors that are fixed in the body frame. This school would answer your question "Yes, the angular velocity of the rigid body is the same in both frames."
The other school could be called the Angular Velocity Addition Theorem school, or Kane's school. In this school one talks about the angular velocity of rigid body B as measured in a frame attached to rigid body A. Since this school distinguishes which body is being measured and which frame it is being measured in, it uses the symbol $^A\omega^B$ to indicate the angular velocity of body B in frame A. The angular velocity can be expressed in terms of basis vectors that are fixed in A, fixed B, fixed in an auxiliary frame or combinations of frames. This school would say the angular velocity of body B in frame B is always zero, because to them the phrase "angular velocity in frame B" means as measured in frame B. The two schools would agree that "the angular velocity of body B as measured in body A is the same no matter what basis vectors are used."
The names I have given these schools reflect how I keep them separate in my mind. When I read Classical Mechanics by Herbert Goldstein, I think in terms of an inertial frame. When I read Dynamics: Theory and Applications or Spacecraft Dyanmics by Thomas Kane et al, I think of adding angular velocity vectors.