First you need to decouple your MIMO model into several SISO loops, each being controlled by one PID controller (using RGA methods for example). Then, depending on the PID tuning method, you need to compute the corresponding model properties (using step response, frequency response, etc). 

Ultimate gains and ultimate frequencies can be computed for a SISO model following their definitions. MapleSim Control Design Toolbox has commands to do that automatically, as well. 

Overall, I agree with Masoud that a simplifcation to a SISO system is usually convenient.

However, Nyquist and root locus stability methods are also available in the case that you system is a MIMO with non-zero terms out of the main diagonal. The big drawback of these system is that the characteristic equation is of high order so the analysys becomes complex. However, if you have a good model of the system you can take advantage of software tools, such as matlab.

As an example, in the following paper

Freijedo et al., "Tuning of Synchronous-Frame PI Current Controllers in Grid-Connected Converters Operating at a Low Sampling Rate by MIMO Root Locus", IEEE TIE, 2015

we calculate/draw the closed loop dominant poles as a function of the control bandwidth (a gain of the PI). Then, from the model we can identify a convenient tuning.

This topic is very well explained in the book by Goodwin et al "Control system Design".

In my opinion, Bode diagrams are not specially good to study stability of MIMO systems.

Robust Bode Methods for Multi–Input/Multi–Output (MIMO) Systems

it is a new graphical control design technique for nonlinear multi–input/multi –output systems based on the Contoured Robust Controller Bode (CRCBode) plot which shows contours (level–sets) of a robust metric on the controller Bode magnitude and phase plot

I total agree Francisco Freijedo. Bode diagrams are not specially good to study  the stability of MIMO systems.

Please refer to these articles:

https://www.researchgate.net/publication/313576646_From_PID_to_Nonlinear_State_Error_Feedback_Controller

https://www.researchgate.net/publication/312046377_Improved_Sliding_Mode_Nonlinear_Extended_State_Observer_based_Active_Disturbance_Rejection_Control_for_Uncertain_Systems_with_Unknown_Total_Disturbance

https://www.researchgate.net/publication/309655805_On_the_Improved_Nonlinear_Tracking_Differentiator_based_Nonlinear_PID_Controller_Design

Article From PID to Nonlinear State Error Feedback Controller

Article Improved Sliding Mode Nonlinear Extended State Observer base...

Article On the Improved Nonlinear Tracking Differentiator based Nonl...

Please refer to this article:

https://www.researchgate.net/publication/318726178_An_Improved_Active_Disturbance_Rejection_Control_for_a_Differential_Drive_Mobile_Robot_with_Mismatched_Disturbances_and_Uncertainties

Conference Paper An Improved Active Disturbance Rejection Control for a Diffe...