Unfortunately, this is a feature Monaco editor does not support. unless Microsoft add code lens/Syntax checking etc to embedded code blocks or Vue add a language parser, this isn't gonna happen any time soon.
You can always fire up vscode with the volar plugin or write your templates in the Vuetify playground
Having just wrote the above, I realise the playground is web based and it supports embedded scripts parsing. I wonder if vuetify have written a language parser that's open to the public? 
UPDATE (and note to self) - there is a volar vue web worker here: GitHub - Kingwl/monaco-volar
