The question is : Why we need plating gold for inner layers ,Will it influence high density Circuit design ?
I believe the short of it is that it's per clients requirement as you stated... Now the question of why they want that is likely best left to them, it could be for something as simple as an advertising gimmick, all the way to an anal engineer that feels it's necessary for whatever reason, and some where in the middle where they have a valid concern based on whatever criteria...
IMO it's likely overkill but they might very well have a valid concern... IMO you generally gold plate to slow down oxidation and corrosion and provide a nice conductive surface, but I don't see much need for this while encapsulated in the PC board substrate... I'm also of the opinion that no matter how strong you make the chain the weakest link is the failure point, thus you can gold plate all this stuff but if you use traditional solder and components that are not gold you are right back to page one with the weak link...
But, those are my opinions not your clients...
As for how to ensure a reliable manufacturing process, that is really a loaded question... The short of it you have to trust your manufacture to do a good job and follow your specifications... Or you can invest a ton of money and resources into taking the product apart and verifying they did a what they where supposed to...