What is this carpet stuff on the dash?

Yep , presumably to protect the underlying material.

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I have one of those to protect the dash from the sun and yes it is aftermarket. They come with small sections of Velcro but I wouldn't want anything sticky on the dash so it just sits and stays in place with no fastener pretty well (slides back if you are really getting on it).

 

My porsche has a custom cut dash cover like that. To avoid sticky I put safety pins underneath cover by the vents with looped fishing line through the vent grill to hold it in place.

You can also use a steamer on the corners that fold down to get it molded better where it sticks up.

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I've not seen that for the 355 before. Would prefer to let the dash get ruined and repair it vs that, myself. I do want a bolster Cavallino though

 

It's quite possible the dash leather has already shrunk and the "protective" carpet is there to fix/hide the damage too.