The instantaneous rotation rate is nothing but the angular velocity of your system. The first who derived the mathematical formulation for the orientation of ellipsoidal bodies is Jeffery and his work dated back to the twenties. He applied the so-called the slender body theory where matching asymptotic solutions could be derived. In his work, the thermal fluctuation has been neglected.

The rotation rate is a quantitative measure of the change in time of the orientation vector. You can define it for a single particle, for a pair or particles or for a bunch of particles knowing the instantaneous positions and velocities. Its unit is thus radians per second.

Alix, you can find your answer in a book called "Eng. Mechanics dynamics" by Meriam with great explanation. I have the book. let me know if you want that.

The value of the angular velocity vector omega(t) at time t ? If so, the unit is Hz or, equivalently, radians per second.

Cartan has a nice treatment of this in terms of the rotation rate of a system of unit vectors that define a system of coordinates that rotate with a rigid body, often called "body axes". Denote the unit vectors of this coordinate system by e_i. Then d e_i / dt = {\omega^k}_i e_k, where {\omega}^k}_i is a mixed rank 2 tensor. Its covariant components \omega_{ki} are antisymmetric, and are the natural description of "angular velocity" - the usual angular velocity vector is not a natural description, being a pseudo-vector constructed from this rank 2 tensor by setting \omega^i = - V^{ijk} \omega_{jk} / 2. Here, the rank three tensor V^{ijk} is the contravariant form of the volume tensor, given by V^{ijk} = (1/\sqrt{ det g } \varepsilon^{ijk}, where det g is the determinant of the (covariant) metric tensor g_{ij} = e_i \cdot e_j, and \varepsilon^{ijk} is the Levi-Civita permutation symbol, sometimes erroneously called a tensor. (It looks like a tensor if your restrict transformations to rotations, but is in reality an instruction for permutation.) 

Brief inspection of the paper you quote (I would have to read it in detail to be certain) suggests that you are dealing with the antisymmetric part of the deformation tensor describing either a flow or the deformation of an elastic solid. The deformation tensor is the Jacobian matrix of the velocity field (in the case of a flow) or of the displacement field (in the case of an elastic deformation). Thus, for a flow, it is the tensor v_{i;j}, where v_i is the covariant velocity field, and ;j indicates the covariant derivative with respect to x^j (same as a partial derivative in Euclidean coordinates). Then \omega_{ij} = v_{ij} - v_{ji}. To say that this is the instantaneous velocity of a pure shear is to say that v_[ij} has zero contraction (trace), that is, {v^i}_i =0. In this case, \omega_{ij} represents the instantaneous angular velocity of a frame consisting of three vectors e_i that co-rotate with an infinitesimal fluid element as the fluid flows. If you construct the associated angular velocity vector (formula given above) the direction of the vector is the axis of rotation of this element at time $t$, while the magnitude of the vector is its angular velocity of rotation about this axis at time t.

These concepts are transferable, mutatis mutandi, to the case of elastic deformation of solid media.