Skeuomorphism is not "old fashioned", It's just no longer as "trendy". However, it is still very relevant.
There is a world of difference between "old fashioned' and "trendy".
The reality is skeuomorphism is still used, but in more subtle ways than previous iterations. Google's Material Design with the long shadows and ripple effects are skeuomorphism traits. In addition, gradients are a staple of all current trends... flat, material, etc. They are merely more subtle gradients than gradients used 10 years ago. That doesn't make them any less skeuomorphic.
Dismissing an entire design moment as "old fashioned" really only means you're set on following trends. Design has always been rooted in history. I have no clue where you get the impression that only Material Design is "modern". It's a trend like any other trend. It's not the only design style in existence. Pigeonholing into one style just means you'll be the last person standing in front of Blockbuster video waiting to rent a VHS tape while everyone else is streaming Netflix.
As for that particular site, my personal opinion is that I would not refer to it as skeuomorphic. Other than the gradients in the header, I see no indication of other skeuomorphic characteristics. And, as I've posted, gradients are still a staple of all current trends as far as I'm aware.
As for "copying the site". My experience tells me when a client requests something like this they are essentially telling you "I like this design style". Naturally you'd have to have a conversations to confirm this. However, generally they can't pinpoint what specifically speaks to them, only that they like the "mood" of the design. It's generally the "mood" I shoot for rather than replicating specific aspects directly.