Seems to me that what your client is trying to do is to manipulate the volume, quality and relevancy of links pointing to the main (mother) website and to try to gain some value for keyphrases within the domain names.
Having the same navigational system may hurt the effort only because of the large number of times he wishes to duplicate it. If it was three or four times it would be safer.
The question about whether this is greyhat, blackhat or whatever is answered when you wrote "...filled with different content...". By different, it cannot be just replicated, mix mashed same old stuff. it must be indisputably unique content, maybe all saying the same stuff but in substantially innovative ways.
So, in spite of the navigational system/linking structure being identical and probably overdone (20 domains) and because you are implying that the content will be different, it should pass the search engine as not being mirrored sites but you'll know and I'll know and the client will know that in fact it is.
Judgement by the court: grey hat. Good luck with it though and try to convince the client to let you recreate each domain using a unique template and here's a great opportunity to try out various manners to throw the links around.
Go to my blog and read up on "Keyphrase Dynamicability" and "Anchor Congruency".