riotjs .tag syntax highlighting

No, there is no way to do this. Please feel free to file a request for Riot framework support to youtrack (
https://youtrack.jetbrains.com/issues/WEB) to let other users vote for it, so that we can see the actual demand for this feature.

just keep your custom tags in html files and use "--ext html" to compile them

<custom-tag>
     <p>Something great<p>

     <script>
          var self = this;
     </script>
</custom-tag>

Or, you could assign the *.tag extension to HTML in Settings->Editor->File Types 

Select HTML from the "Recognized File Types" list (top panel), then add a new item in the "Registered Patterns" list (the second panel) by clicking the "+" button on the right (in the second panel).