Why does a model that is used for policy analysis needs its shocks and parameters to be structural (= invariant) to the shock under consideration?
This is due to the famous Lucas critique. 

To make long story short, in the past in the heyday of Keynesian macroeconomics it was quite normal for macroeconomists to just postulate some relationships based on relatively casual empirical observations like for example the Philips curve which says that there is positive relationship between inflation and employment. 

However, this ignores some fundamental facts that such relationship only holds as long as people do not expect inflation. The (original) version of Philips curve did not really postulate any deep relationship between the two variables just assumed that it holds.

In short Lucas harshly criticized such approach as the (old) Keynesian approach gives you the correct predictions only as long as you assume that people are passive agents that do not adjust their behavior when government changes its policy. But that is not really reasonable assumption as we know from microeconomics people are not just passive actors, when government changes its policy people try to find ways how to take the maximum advantage of the new policy regime. 

The Lucas critique is basically a generalisation of the Goodhart's law which states that "when a measure becomes a target, it ceases to be a good measure"

If you add structural microfoundations you create more rich model which actually has an ability to take account for peoples reaction to changes in policy.

For example, the New Keynesian version of Philips curve instead of assuming that there is positive relationship explicitly models peoples/firms expectations and shows that when  they do not expect inflation it is rational for them to increase employment but when they do it is rational for them to just adjust contracts in a way to account for inflation (for example linking pays to inflation). Hence in such model no matter what is the government policy (with respect to inflation) the answer will be always 'correct' in the sense that it does not just assumes there is a relationship but it explicitly models the reasons/mechanisms why there is sometimes positive and sometimes no relationship between inflation and employment. 


  Isn't it reasonable to assume that, in addition to many economic variables, also many parameters that govern micro behaviour will be influenced by a policy shock?


Not really in the sense the microfoundations are set in the New Keynesian macro models. In such models when we talk about microfoundations we talk about proving the results of the model assuming that for example people are rational, have rational expectations and interact in the markets. 

To affect such micro behavior the policy would have actually change fundamentally how people behave. 

I suppose from sociological perspective you could argue that in a long term policy can change how people behave (I am not expert on that). But from economic perspective it is completely valid to say that model with solid microfoundations is policy invariant. 

Moreover, note an extremely important caveat is what the authors from which you quote say here: 


  Must be invariant with respect to the policy interventions considered.


Hence, it is not even necessary to make structural model that is invariant to any conceivable policy that you can imagine but rather the policies that are important for the model, like in Philips curve transitioning from market economy to centrally planned economy where property rights are abolished would certainly invalidate many micro foundations and probably changed conclusions, but different monetary policy doesn't and thats what is important. 



Regarding your edit of a question, when it comes to microfoundations its not really about rational/adaptive expectations per se - it is about justifying people having such expectations based on some fundamental model of human behavior.  

De Grauwe (and others) in his recent work takes a lot of inspiration from behavioral economics. In a sense his models are microfounded they just tweak the classical rational choice theory by adding some behavioral flavor to it. But I used the word in 'a sense' on purpose because to my best knowledge there is no general behavioral theory of human behavior. Rather he takes the rational choice theory and tweaks it by assuming that people are sometimes rational but in other situations not (or many of his models can be also interpreted as saying some portion of population is rational while other not). 

Hence he uses some sort of microfoundations in a way that he has some model of fundamental human behavior based on which results are derived. But I would say there is some room for disagreement about that because since there is no general behavioral theory (at least not one accepted in macro for sure) you could argue that his assumptions on human behavior are ad hoc and not proper microfundations as he just assumes some behavioral regularities without really proving that they are not  policy dependent or giving some ‘deeper’ justification why some behavioral assumptions are made and not other other than showing that there is some support from them from behavioral economics or evidence (although I am just playing devils advocate - I actually really like the work of De Grauwe).