How can I make a VSCode theme recognize C# interfaces?

editor.tokenColorCustomizations can use a number of values: comments, functions, keywords, numbers, strings, types and variables. If none of those work for you textMateRules is available as well. So you can do something like:

"editor.tokenColorCustomizations": {
    "textMateRules": [{
        "scope": "yourScopeHere",
        "settings": {
            "fontStyle": "italic",
            "foreground": "#C69650"
        }
    }]
   },

So you just have to figure out what scope you need for "interface".

For that, try CTRL-Shift-P and type scope: choose

Developer: Inspect TM Scopes  

and for whichever keyword is selected, like interface you will get a listing of its textmate scope. That should be inserted as the scope value above. [In my experience, it is more accurate to open the "Inspect TM Scopes" panel and then click a couple of items and then the one, like interface, that you want - the scope panel will remain open.] You can copy from the scopes panel.

You may need only the main scope listed, but if need to narrow its scope you can include the others listed in a comma-separated list in the scopes: ..., ..., ...