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I have downloaded the shapefile for European NUTS region fro here.

In particular I have downloaded the 1:1 Million SHP file for NUTS 2021 from here.

I have opened them using QGIS.

In each region there are 3 categorical variables, named "MOUNT_TYPE", "URBN_TYPE", and "COAST_TYPE", as can be seen in this example from Mance:

enter image description here

What does the value of that categorical variables represent?

There is a "metadata.pdf" file together with the shapefile that doesn't explain them. Also the online reference in that PDF file doesn't explain them.

I googled for them but could not find anything.

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Following the link in the metadata file you arrive here where is states:

Specific geographies such as coastal regions, mountain regions, border regions or island regions are also covered.

It's does not take a massive leap in imagine to realise that Mount/UrBn/Coast type fields relate to these typologies.

That said I completely agree their metadata falls short of explaining what these fields are and their values. EUROSTAT appear to have "dropped the ball" on this and published their data without updating their metadata. Would have expected better...

Fortunately for you I clicked on a few more links and ended up here, click on the NUTS 2021 classification link to download an Excel file and the information you seek (what the codes mean) are embedded in the sheets.

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