How can I compare degree of stability (or persistency) in two dynamic systems/models?
The most straightforward way is to examine the coefficients of your VAR model, since the persistence is broadly understood in econometrics as a system dependence on its own past (see Verbeek (2008): a guide to modern econometrics).  A simple VAR model is given:
$${\displaystyle y_{1,t}=c_{1}+a_{1,1}y_{1,t-1}+a_{1,2}y_{2,t-1}+e_{1,t}\,} \\
{\displaystyle y_{2,t}=c_{2}+a_{2,1}y_{1,t-1}+a_{2,2}y_{2,t-1}+e_{2,t}.\,}$$
Now, in very similar fashion as with AR models how persistent the variable is will be given by how much it depends on its own past (lags). For example, persistence of $y_1$ can be gauged by coefficient $a_{1,1}$. The larger the coefficient the more persistent the series is.