Is there a smart way to minimize the size tab view

Sorry, there is no support to modify the height. What is your use case?

Have you checked out the other style that Appian provides?  I think it was called Helium.

Client is not impressed with that section, client wants to remove the whole tab section including the profile icon, the second best thing he would settle is minimizing the height.

The only things that can be changed here are those elements which are directly configurable via the Site Branding setup.  Companies that utilize Appian for its quick and flexible design should already be familiar with the fact that certain elements of the design cannot be controlled at a pixel-by-pixel level.

Actually, it's Helium and Mercury.

https://docs.appian.com/suite/help/22.2/ux_site_branding.html

Those are your options.  You can take advantage of the style with no additional pages, so it won't show any tabs at all, it sounds like.  Beyond that, it's just how Appian works.

Part of my daily work is to advice and educate my clients in using Appian in the most efficient and effective way. Pixel perfect layouts is not part of it.